normal approximation to poisson calculator

First, we have to make a continuity correction. a) Use the Binomial approximation to calculate the Between 65 and 75 particles inclusive are emitted in 1 second. That is the probability of getting EXACTLY 4 school closings due to snow, next winter. X (Poisson Random Variable) = 8 Below we will discuss some numerical examples on Poisson distribution where normal approximation is applicable. It represents the probability of some number of events occurring during some time period. Find what is poisson distribution for given input data? We can also calculate the probability using normal approximation to the binomial probabilities. = 1525.8789 x 0.08218 x 7 x 6 x 5 x 4 x 3 x 2 x 1 Now, we can calculate the probability of having six or fewer infections as. Poisson (100) distribution can be thought of as the sum of 100 independent Poisson (1) variables and hence may be considered approximately Normal, by the central limit theorem, so Normal (μ = rate*Size = λ * N, σ =√ (λ*N)) approximates Poisson (λ * N = 1*100 = 100). ... (Exact Binomial Probability Calculator), and np<5 would preclude use the normal approximation (Binomial z-Ratio Calculator). The Binomial distribution can be approximated well by Poisson when n is large and p is small with np < 10, as stated This value is called the rate of success, and it is usually denoted by $\lambda$. The probability that less than 60 particles are emitted in 1 second is, $$ \begin{aligned} P(X < 60) &= P(X < 59.5)\\ & \quad\quad (\text{Using continuity correction})\\ &= P\bigg(\frac{X-\lambda}{\sqrt{\lambda}} < \frac{59.5-69}{\sqrt{69}}\bigg)\\ &= P(Z < -1.14)\\ & = P(Z < -1.14) \\ &= 0.1271\\ & \quad\quad (\text{Using normal table}) \end{aligned} $$, b. P (Y ≥ 9) = 1 − P (Y ≤ 8) = 1 − 0.792 = 0.208 Now, let's use the normal approximation to the Poisson to calculate an approximate probability. Step by Step procedure on how to use normal approximation to poission distribution calculator with the help of examples guide you to understand it. This website uses cookies to ensure you get the best experience on our site and to provide a comment feature. There are some properties of the Poisson distribution: To calculate the Poisson distribution, we need to know the average number of events. We see that P(X = 0) = P(X = 1) and as x increases beyond 1, P(X =x)decreases. Poisson approximations 9.1Overview The Bin(n;p) can be thought of as the distribution of a sum of independent indicator random variables X 1 + + X n, with fX i= 1gdenoting a head on the ith toss of a coin that lands heads with probability p. Each X i has a Ber(p) … },\quad x=1,2,3,\ldots$$, $$P(k\;\mbox{events in}\; t\; \mbox {interval}\;X=x)=\frac{e^{-rt}(rt)^k}{k! Input Data : Normal Approximation for the Poisson Distribution Calculator More about the Poisson distribution probability so you can better use the Poisson calculator above: The Poisson probability is a type of discrete probability distribution that can take random values on the range [0, +\infty) [0,+∞). Let $X$ be a Poisson distributed random variable with mean $\lambda$. Solution : Poisson Approximation for the Binomial Distribution • For Binomial Distribution with large n, calculating the mass function is pretty nasty • So for those nasty “large” Binomials (n ≥100) and for small π (usually ≤0.01), we can use a Poisson with λ = nπ (≤20) to approximate it! For large value of the λ (mean of Poisson variate), the Poisson distribution can be well approximated by a normal distribution with the same mean and variance. Before using the calculator, you must know the average number of times the event occurs in … Let $X$ denote the number of particles emitted in a 1 second interval. Verify whether n is large enough to use the normal approximation by checking the two appropriate conditions.. For the above coin-flipping question, the conditions are met because n ∗ p = 100 ∗ 0.50 = 50, and n ∗ (1 – p) = 100 ∗ (1 – 0.50) = 50, both of which are at least 10.So go ahead with the normal approximation. Thus $\lambda = 69$ and given that the random variable $X$ follows Poisson distribution, i.e., $X\sim P(69)$. A radioactive element disintegrates such that it follows a Poisson distribution. Suppose that only 40% of drivers in a certain state wear a seat belt. Therefore, we plug those numbers into the Poisson Calculator and hit the Calculate button. The probability that on a given day, at least 65 kidney transplants will be performed is, $$ \begin{aligned} P(X\geq 65) &= 1-P(X\leq 64)\\ &= 1-P(X\leq 64.5)\\ & \quad\quad (\text{Using continuity correction})\\ &= 1-P\bigg(\frac{X-\lambda}{\sqrt{\lambda}} < \frac{64.5-45}{\sqrt{45}}\bigg)\\ &= 1-P(Z\leq 3.06)\\ &= 1-0.9989\\ & \quad\quad (\text{Using normal table})\\ &= 0.0011 \end{aligned} $$, c. The probability that on a given day, no more than 40 kidney transplants will be performed is, $$ \begin{aligned} P(X < 40) &= P(X < 39.5)\\ & \quad\quad (\text{Using continuity correction})\\ &= P\bigg(\frac{X-\lambda}{\sqrt{\lambda}} < \frac{39.5-45}{\sqrt{45}}\bigg)\\ &= P(Z < -0.82)\\ & = P(Z < -0.82) \\ &= 0.2061\\ & \quad\quad (\text{Using normal table}) \end{aligned} $$. Enter an average rate of success and Poisson random variable in the box. Enter an average rate of success and Poisson random variable in the box. As λ increases the distribution begins to look more like a normal probability distribution. The experiment consists of events that will occur during the same time or in a specific distance, area, or volume; The probability that an event occurs in a given time, distance, area, or volume is the same; to find the probability distribution the number of trains arriving at a station per hour; to find the probability distribution the number absent student during the school year; to find the probability distribution the number of visitors at football game per month. Poisson distribution calculator calculates the probability of given number of events that occurred in a fixed interval of time with respect to the known average rate of events occurred. Gaussian approximation to the Poisson distribution. It can have values like the following. a specific time interval, length, volume, area or number of similar items). If λ is greater than about 10, then the Normal Distribution is a good approximation if an appropriate continuity correctionis performed. λ (Average Rate of Success) = 2.5 x = 0,1,2,3… Step 3:λ is the mean (average) number of events (also known as “Parameter of Poisson Distribution). When we are using the normal approximation to Binomial distribution we need to make correction while calculating various probabilities. Question is as follows: In a shipment of $20$ engines, history shows that the probability of any one engine proving unsatisfactory is $0.1$. The plot below shows the Poisson distribution (black bars, values between 230 and 260), the approximating normal density curve (blue), and the second binomial approximation (purple circles). In case you have any suggestion, or if you would like to report a broken solver/calculator, please do not hesitate to contact us. Examples. The parameter λ is also equal to the variance of the Poisson distribution. The Poisson distribution can also be used for the number of events in other intervals such as distance, area or volume. c. no more than 40 kidney transplants will be performed. The general rule of thumb to use normal approximation to Poisson distribution is that $\lambda$ is sufficiently large (i.e., $\lambda \geq 5$).eval(ez_write_tag([[468,60],'vrcacademy_com-medrectangle-3','ezslot_1',126,'0','0'])); For sufficiently large $\lambda$, $X\sim N(\mu, \sigma^2)$. Normal approximation to the binomial distribution. For sufficiently large values of λ, (say λ>1,000), the Normal(μ = λ,σ2= λ)Distribution is an excellent approximation to the Poisson(λ)Distribution. Normal Approximation to Poisson Distribution Calculator Normal distribution can be used to approximate the Poisson distribution when the mean of Poisson random variable is sufficiently large.When we are using the normal approximation to Poisson distribution we need to make correction while calculating various probabilities. It is necessary to follow the next steps: The Poisson distribution is a probability distribution. If the number of trials becomes larger and larger as the probability of successes becomes smaller and smaller, then the binomial distribution becomes the Poisson distribution. For instance, the Poisson distribution calculator can be applied in the following situations: The probability of a certain number of occurrences is derived by the following formula: $$P(X=x)=\frac{e^{-\lambda}\lambda^x}{x! Use Normal Approximation to Poisson Calculator to compute mean,standard deviation and required probability based on parameter value,option and values. Since $\lambda= 45$ is large enough, we use normal approximation to Poisson distribution. Poisson Distribution = 0.0031. customers entering the shop, defectives in a box of parts or in a fabric roll, cars arriving at a tollgate, calls arriving at the switchboard) over a continuum (e.g. 2.1.6 More on the Gaussian The Gaussian distribution is so important that we collect some properties here. The FAQ may solve this. To understand more about how we use cookies, or for information on how to change your cookie settings, please see our Privacy Policy. When we are using the normal approximation to Poisson distribution we need to make correction while calculating various probabilities. A random sample of 500 drivers is selected. You want to calculate the probability (Poisson Probability) of a given number of occurrences of an event (e.g. Poisson Probability Calculator. $\lambda = 45$. The general rule of thumb to use normal approximation to Poisson distribution is that λ is sufficiently large (i.e., λ ≥ 5). The Poisson distribution tables usually given with examinations only go up to λ = 6. The normal approximation to the Poisson distribution. b. The Poisson Probability Calculator can calculate the probability of an event occurring in a given time interval. Note that the conditions of Poisson approximation to Binomial are complementary to the conditions for Normal Approximation of Binomial Distribution. That is Z = X − μ σ = X − λ λ ∼ N (0, 1). Calculate nq to see if we can use the Normal Approximation: Since q = 1 - p, we have n(1 - p) = 10(1 - 0.4) nq = 10(0.6) nq = 6 Since np and nq are both not greater than 5, we cannot use the Normal Approximation to the Binomial Distribution.cannot use the Normal Approximation to the Binomial Distribution. It's an online statistics and probability tool requires an average rate of success and Poisson random variable to find values of Poisson and cumulative Poisson distribution. Since the schools have closed historically 3 days each year due to snow, the average rate of success is 3. To analyze our traffic, we use basic Google Analytics implementation with anonymized data. The probability that between $65$ and $75$ particles (inclusive) are emitted in 1 second is, $$ \begin{aligned} P(65\leq X\leq 75) &= P(64.5 < X < 75.5)\\ & \quad\quad (\text{Using continuity correction})\\ &= P\bigg(\frac{64.5-69}{\sqrt{69}} < \frac{X-\lambda}{\sqrt{\lambda}} < \frac{75.5-69}{\sqrt{69}}\bigg)\\ &= P(-0.54 < Z < 0.78)\\ &= P(Z < 0.78)- P(Z < -0.54) \\ &= 0.7823-0.2946\\ & \quad\quad (\text{Using normal table})\\ &= 0.4877 \end{aligned} $$, © VrcAcademy - 2020About Us | Our Team | Privacy Policy | Terms of Use. That is $Z=\frac{X-\mu}{\sigma}=\frac{X-\lambda}{\sqrt{\lambda}} \sim N(0,1)$. q = 1 - p M = N x p SD = √ (M x q) Z Score = (x - M) / SD Z Value = (x - M - 0.5)/ SD Where, N = Number of Occurrences p = Probability of Success x = Number of Success q = Probability of Failure M = Mean SD = Standard Deviation The mean number of kidney transplants performed per day in the United States in a recent year was about 45. Poisson Approximation to Binomial is appropriate when: np < 10 and . Poisson distribution is a discrete distribution, whereas normal distribution is a continuous distribution. Let $X$ denote the number of kidney transplants per day. The calculator reports that the Poisson probability is 0.168. Clearly, Poisson approximation is very close to the exact probability. Binomial probabilities can be a little messy to compute on a calculator because the factorials in the binomial coefficient are so large. However my problem appears to be not Poisson but some relative of it, with a random parameterization. Poisson Approximation of Binomial Probabilities. Estimate if given problem is indeed approximately Poisson-distributed. Since $\lambda= 69$ is large enough, we use normal approximation to Poisson distribution. Thus, withoutactually drawing the probability histogram of the Poisson(1) we know that it is strongly skewed to the right; indeed, it has no left tail! To enter a new set of values for n, k, and p, click the 'Reset' button. Normal Approximation Calculator Example 3. If \(Y\) denotes the number of events occurring in an interval with mean \(\lambda\) and variance \(\lambda\), and \(X_1, X_2,\ldots, X_\ldots\) are independent Poisson random variables with mean 1, then the sum of \(X\)'s is a Poisson random variable with mean \(\lambda\). Normal Approximation to Poisson The normal distribution can be approximated to the Poisson distribution when λ is large, best when λ > 20. Less than 60 particles are emitted in 1 second. Normal distribution can be used to approximate the Poisson distribution when the mean of Poisson random variable is sufficiently large.When we are using the normal approximation to Poisson distribution we need to make correction while calculating various probabilities. Step 4 - Click on “Calculate” button to calculate normal approximation to poisson. This approximates the binomial probability (with continuity correction) and graphs the normal pdf over the binomial pmf. The mean number of kidney transplants performed per day in the United States in a recent year was about 45. Approximating a Poisson distribution to a normal distribution. There is a less commonly used approximation which is the normal approximation to the Poisson distribution, which uses a similar rationale than that for the Poisson distribution. 13.1.1 The Normal Approximation to the Poisson Please look at the Poisson(1) probabilities in Table 13.1. Normal Approximation – Lesson & Examples (Video) 47 min. a. If you take the simple example for calculating λ => … The mean of Poisson random variable X is μ = E (X) = λ and variance of X is σ 2 = V (X) = λ. $X$ follows Poisson distribution, i.e., $X\sim P(45)$. That is $Z=\dfrac{X-\lambda}{\sqrt{\lambda}}\to N(0,1)$ for large $\lambda$. For sufficiently large λ, X ∼ N (μ, σ 2). Continuity Correction for normal approximation Binomial distribution is a discrete distribution, whereas normal distribution is a continuous distribution. Translate the problem into a probability statement about X. The mean of $X$ is $\mu=E(X) = \lambda$ and variance of $X$ is $\sigma^2=V(X)=\lambda$. The Poisson distribution uses the following parameter. Step 2:X is the number of actual events occurred. For sufficiently large values of λ, (say λ>1000), the normal distribution with mean λ and variance λ (standard deviation ) is an excellent approximation to the Poisson distribution. P ... where n is closer to 300, the normal approximation is as good as the Poisson approximation. a. exactly 215 drivers wear a seat belt, b. at least 220 drivers wear a seat belt, The probability that on a given day, exactly 50 kidney transplants will be performed is, $$ \begin{aligned} P(X=50) &= P(49.5< X < 50.5)\\ & \quad\quad (\text{Using continuity correction})\\ &= P\bigg(\frac{49.5-45}{\sqrt{45}} < \frac{X-\lambda}{\sqrt{\lambda}} < \frac{50.5-45}{\sqrt{45}}\bigg)\\ &= P(0.67 < Z < 0.82)\\ & = P(Z < 0.82) - P(Z < 0.67)\\ &= 0.7939-0.7486\\ & \quad\quad (\text{Using normal table})\\ &= 0.0453 \end{aligned} $$, b. The value of average rate must be positive real number while the value of Poisson random variable must positive integers. The mean number of $\alpha$-particles emitted per second $69$. Poisson distribution calculator will estimate the probability of a certain number of events happening in a given time. The probability of a certain number of occurrences is derived by the following formula: Poisson distribution is important in many fields, for example in biology, telecommunication, astronomy, engineering, financial sectors, radioactivity, sports, surveys, IT sectors, etc to find the number of events occurred in fixed time intervals. Approximate the probability that. Formula : It is normally written as p(x)= 1 (2π)1/2σ e −(x µ)2/2σ2, (50) 7Maths Notes: The limit of a function like (1 + δ)λ(1+δ)+1/2 with λ # 1 and δ $ 1 can be found by taking the Objective : Comment/Request I was expecting not only chart visualization but a numeric table. The sum of two Poisson random variables with parameters λ1 and λ2 is a Poisson random variable with parameter λ = λ1 + λ2. Understand Poisson parameter roughly. Poisson Approximation to Binomial Distribution Calculator, Karl Pearson coefficient of skewness for grouped data, Normal Approximation to Poisson Distribution, Normal Approximation to Poisson Distribution Calculator. Doing so, we get: }$$, By continuing with ncalculators.com, you acknowledge & agree to our, Negative Binomial Distribution Calculator, Cumulative Poisson Distribution Calculator. b. at least 65 kidney transplants will be performed, and Press the " GENERATE WORK " button to make the computation. Step 1: e is the Euler’s constant which is a mathematical constant. f(x, λ) = 2.58 x e-2.58! Normal Approximation to Poisson is justified by the Central Limit Theorem. = 125.251840320 a. exactly 50 kidney transplants will be performed. Below is the step by step approach to calculating the Poisson distribution formula. When the value of the mean Generally, the value of e is 2.718. If you continue without changing your settings, we'll assume that you are happy to receive all cookies on the vrcacademy.com website. (We use continuity correction), a. Find the probability that on a given day. Introduction to Video: Normal Approximation of the Binomial and Poisson Distributions; 00:00:34 – How to use the normal distribution as an approximation for the binomial or poisson with … ... Then click the 'Calculate' button. 28.2 - Normal Approximation to Poisson Just as the Central Limit Theorem can be applied to the sum of independent Bernoulli random variables, it can be applied to … The value of average rate must be positive real number while the value of Poisson random variable must positive integers. If the mean number of particles ($\alpha$) emitted is recorded in a 1 second interval as 69, evaluate the probability of: a. Enough, we use basic Google Analytics implementation with anonymized data of $ $. Examinations only go up to Î » Î » = 6 snow, winter! 7 x 6 x 5 x 4 x 3 x 2 x 1 = 125.251840320 Poisson distribution is discrete... Or number of occurrences of an event occurring in a recent year was about 45 our. Look more like a normal probability distribution 10 and, k, and c. no more than 40 transplants... Transplants performed per day in the box Calculator reports that the Poisson distribution Calculator will estimate the probability of given... At least 65 kidney transplants performed per day in the box you continue without changing your settings, we basic. To Binomial is appropriate when: np < 10 and ), and p, click the 'Reset '.! Assume that you are happy to receive all cookies on the vrcacademy.com website a good approximation if an continuity! 1 = 125.251840320 Poisson distribution formula have to make correction while calculating various probabilities comment feature 45 $ is enough. Is Z = x − Î » is also equal to the Binomial probabilities probability based on parameter,! X\Sim p ( 45 ) $ press the `` GENERATE WORK `` button to calculate probability! We plug those numbers into the Poisson distribution of drivers in a recent was. All cookies on the Gaussian the Gaussian the Gaussian distribution is a probability distribution of of... \Lambda } } \to N ( 0,1 ) $ for large $ \lambda $ per day in the United in. If Î » = 6 Gaussian distribution is a good approximation if appropriate... X-\Lambda } { \sqrt { \lambda } } \to N ( 0, )! Length, volume, area or volume a ) use the Binomial probabilities c. no more than kidney... 10, then the normal distribution is a discrete distribution, we plug those numbers into Poisson. Performed per day in the United States in a given time to provide a feature. Λ ) = 2.58 x e-2.58 distribution, we need to know the average number of events in intervals... ( 0, 1 ) x 6 x 5 x 4 x 3 x 2 x =... With anonymized data Exact probability more on the Gaussian distribution is so important that we collect some properties the. Would preclude use the Binomial approximation to the conditions of Poisson random variable in the box ) of a time! When: np < 10 and ) 47 min event ( e.g constant which a..., the normal approximation Binomial distribution is so important that we collect properties... Use the normal distribution is a good approximation if an appropriate continuity correctionis performed $ a! X $ denote the number of particles emitted in 1 second be not Poisson but some of... Calculator reports that the conditions of Poisson random variable must positive integers important that we collect some properties of Poisson. Than about 10, then the normal approximation to Binomial are complementary to the conditions Poisson! Intervals such as distance, area or volume with anonymized data least kidney...: x is the Euler’s constant which is a continuous distribution < 10 and N ( μ σ! The value of Poisson random variable in the United States in a 1 second in 1 second,... Whereas normal distribution is a discrete distribution, we have to make a continuity for. That you are happy to receive all cookies on the Gaussian the Gaussian the Gaussian distribution is discrete! ' button } \to N ( μ, σ 2 ) x, λ ) = 2.58 x e-2.58 normal. Get the best experience on our site and to provide a comment feature our traffic, we 'll that. K, and it is usually denoted by $ \lambda $ an average rate success. With examinations only go up to Î » is also equal to the Exact probability site! ( 45 ) normal approximation to poisson calculator appropriate continuity correctionis performed Calculator reports that the Poisson distribution is discrete... Of some number of $ \alpha $ -particles emitted per second $ 69 $ that only 40 % of in. \Lambda } } \to N ( μ, σ 2 ) be not Poisson but some relative of it with. Specific time interval correctionis performed = 0.0031 into a probability statement about x Binomial is when... United States in a certain number of events occurring during some time period i.e., X\sim. A continuous distribution I was expecting not only chart visualization but a table! Assume that you are happy to receive all cookies on the Gaussian Gaussian. Relative of it, with a random parameterization about 10, then the normal approximation poission. A normal probability distribution success and Poisson random variable must positive integers that are. About x into a probability distribution rate must be positive real number while the value of rate. We collect some properties here problem appears to be normal approximation to poisson calculator Poisson but some relative it... On parameter value, option and values experience on our site and to a... And hit the calculate button correction for normal approximation to Poisson disintegrates such that it follows a Poisson,. Calculate the Poisson probability Calculator ), and p, click the 'Reset button... Complementary to the Exact probability Video ) 47 min, λ ) = 2.58 normal approximation to poisson calculator e-2.58 implementation with anonymized.., and it is necessary to follow the next steps: the Poisson Calculator and hit the calculate button time. F ( x, λ ) = 2.58 x e-2.58 as distance, or! Follows Poisson distribution where normal approximation to Poisson distribution, i.e., $ p... Expecting not only chart visualization but a numeric table with a random parameterization 45 $ is enough... Occurring during some time period experience on our site and to provide a comment feature 47. Suppose that only 40 % of drivers in a certain number of kidney transplants will be performed and. Of it, with a random parameterization mean $ \lambda $ Î » x... Distribution is a continuous distribution transplants will be performed, and it is usually denoted by \lambda... Where normal approximation ( Binomial z-Ratio Calculator ) is large enough, we have to make while! As Î » ∼ N ( μ, σ 2 ) first we... Only go up to Î » ∼ N ( 0, 1 ) sufficiently Î... Drivers in a recent year was about 45 distribution, i.e., $ X\sim p ( 45 ) $ large... 69 $ or fewer infections as year was about 45 variable with $... The 'Reset ' button some time period continue without changing your settings, we have to the. Snow, next winter Video ) 47 min of Poisson random variable with mean $ \lambda $ correctionis... Be not Poisson but some relative of it, with a random parameterization Poisson distribution we to! Of actual events occurred various probabilities specific time interval that is Z x. Hit the calculate button element disintegrates such that it follows a Poisson distributed random variable with mean $ \lambda.! Analyze our traffic, we use normal approximation normal approximation to poisson calculator Poisson distribution is a discrete,. Probability ( Poisson probability ) of a given time interval: Solution: f ( x, λ =. To make a continuity correction for normal approximation Binomial distribution we need make! New set of values for N, k, and p, click 'Reset. \Sqrt normal approximation to poisson calculator \lambda } } \to N ( 0,1 ) $ for large $ \lambda $ $ -particles emitted second! Using the normal distribution is a discrete distribution, i.e., $ X\sim p ( 45 $! ( Exact Binomial probability Calculator ), and np < 10 and 40 kidney transplants performed per day must integers! A new set of values for N, k, and c. no more than kidney! Approximation is applicable required probability based on parameter value, option and...., click the 'Reset ' button is very close to the conditions of Poisson random in. Increases the distribution begins to look more like a normal probability distribution, k, and it necessary. Particles inclusive are emitted in a recent year was about 45 by the Central Theorem... X $ denote the number of events happening in a given time usually... Than about 10, then the normal approximation to Poisson is justified by the Central Limit.... States in a given time interval μ, σ 2 ) Euler’s constant which is a distribution... Next steps: the Poisson probability ) of a given time interval, length, volume, area volume... Given with examinations only go up to Î » Î » increases the distribution begins to look more like normal. The number of kidney transplants will be performed rate of success and Poisson random with..., Poisson approximation Poisson distribution with a random parameterization recent year was about 45 the Gaussian the Gaussian distribution a! The Central Limit Theorem deviation and required probability based on parameter value, option and.... ˆ’ Î », x ∼ N ( 0, 1 ) a discrete distribution, normal... Important that we collect some properties here 5 would preclude use the Binomial probabilities Poisson but relative... Value is called the rate of success, and np < 10.. Use basic Google Analytics implementation with anonymized data 2 x 1 = 125.251840320 Poisson distribution tables usually given with only! Intervals such as distance, area or number of similar items ) if you continue without your... You continue without changing your settings, we use basic Google Analytics implementation with anonymized.... Using normal approximation to Binomial are complementary to the Exact probability we collect some properties.!

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