Let X be a random variable. (a) If X~ Binom(n, 0.3) for large n, find an interval in which roughly 68% of the values of X lie. (b) If instead X ~ Pois (20000), assuming that this can be modeled by a normal random variable find an interval in which roughly 68% of the values of X lie.

Let X be a random variable. (a) If X~ Binom(n, 0.3) for large n, find an interval in which roughly 68% of the values of X lie. (b) If instead X ~ Pois (20000), assuming that this can be modeled by a normal random variable find an interval in which roughly 68% of the values of X lie.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

Let X be a random variable.
(a) If X ~ Binom(n, 0.3) for large n, find an interval in which roughly 68% of the values of X lie.
(b) If instead X~ Pois (20000), assuming that this can be modeled by a normal random variable find
an interval in which roughly 68% of the values of X lie.

Expert Solution

Step 1: Determine an interval for large n

(a) For a binomial random variable with parameters n and p :
Mean, $mu equals n p$
Variance, $Var left parenthesis X right parenthesis equals n p left parenthesis 1 minus p right parenthesis$
Standard deviation, $sigma equals square root of n p left parenthesis 1 minus p right parenthesis end root$
Given n is large and p=0.3,

$mu equals n cross times 0.3 sigma equals square root of n cross times 0.3 cross times 0.7 end root$

For roughly $68 percent sign$ of the data in a normal distribution:
Interval is $mu plus-or-minus sigma$
Thus, the interval is:

$n cross times 0.3 minus square root of n cross times 0.3 cross times 0.7 end root comma n cross times 0.3 plus square root of n cross times 0.3 cross times 0.7 end root$

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