Question

STA 2171 HW3 Page 3 of 12 2. We know under certain conditions that the Normal distribution approximates the Bino- mial distribution. But the Normal distribution approximates another important discrete distribution: the Poisson distribution. The Poisson distribution is a one-parameter distribution frequently used to model the number of events that occur over a specified time period. For example, if we are interested in the modeling the number of babies born in a particular hospital during one day, then we might consider this count of new borns as a Poisson(X) random variable, where is the average number of babies born in this hospital per day. Some useful facts about Poisson random variables are: • If X Poisson(X), then E(X) = ax = and Var(X) = o} = X. That is, the mean and variance are the same. . If X Poisson(A) and Y Poisson() are independent, then X+Y~ Poisson( + ) If X Poisson(X) and is large, then the Normal approximation with the continuity correction since the Poisson distribution is discrete) is P(X <x) = P(Y <x+0.5) where Y N (x,c) with Mix the mean of X and of the variance of X. Suppose X Poisson (60). Answer the following. (a) (3 points) Find ux, the mean of X, and o, the variance of X. (b) (3 points) Use the Normal approximation with the continuity correction to find P(58 < X < 62). Show your work, including any calculator functions you use, and round your final answer to 5 decimal places.

Answer 1

Suppose X ~ Poisson (60) where

09 =

(a) The mean of X is

A = A = 60

The variance of X is

= 1 = 60

(b) Use the normal approximation with the continuity correction, the probability is described as

P(58 < X < 62) = P(58 -0.5 <Y < 62 +0.5)

P(58 < X < 62) = P(57.5 <Y < 62.5)

Under central limit theorem,

P(57.5 <Y < 62,5) = P(57.5 - 60 P(57.5 Y 02.0) = Y - Hr62.5 - 60 0 60 60 -

P(57.5 <Y < 62.5) = P(-0.3227 < Z < 0.3227)

P(57.5 <Y < 62.5) = P(Z < 0.3227) - P(Z < -0.3227)

P(57.5 <Y < 62.5) = 0 (0.3227) - o(-0.3227)

Using standard normal table,

P(57.5 <Y < 62.5) = 0.6265 – 0.3734 = 0.2531