1. Suppose that a box contains 30 blue blocks and 20 yellow blocks, and we rando

Question

1. Suppose that a box contains 30 blue blocks and 20 yellow blocks, and we randomly pick 5 blocks from the box. Let X be the random variable denoting the number of blue blocks that are picked. Consider the following two situations: (a) We choose the 5 blocks with replacement (i) Does X follow a Binomial or a Hypergeometric distribution? ii) What is the probability mass function (PMF) for X? iii) What is the Expected value, Variance, and Standard deviation of X? b) We choose the 5 blocks without replacement. (i) Does X follow a Binomial or a Hypergeometric distribution? (ii) What is the probability mass function (PMF) for X? iii) What is the Expected value, Variance, and Standard deviation of X?

Explanation / Answer

A) i) X follows binomial distribution

II) p = 30/50 = 0.6

n = 50

P(x) = 50Cx * 0.6x * 0.450-x

IIi) expected value = n*p = 50 * 0.6 = 30

Variance = n*p*(1-p) = 50 * 0.6 * 0.4 = 12

Standard deviation = sqrt(12) = 3.46

2)i) X follows hypergeometric distribution

II) N = 50

K = 30

n = 5

P(x ) = 30Cx * 20C5-x / 50C5

IIi) expected value = n*k/N = 5*30/50 = 3

Variance = n * k * (N - k) * (N - n) /(N2 (N - 1))

= 5 * 30 * 20 * 45 / (50*50*49)

= 1.1

Standard deviation = sqrt(1.1) = 1.05

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