Question

An overseas shipment of 5 foreign automobiles contains 2 that have slight paint

blemishes. If an agency receives 3 of these automobiles at random, find the

probability distribution of the random variable X representing the number of

automobiles with paint blemishes purchased by the agency. Find the mean

number of automobiles with paint blemishes. Also, calculate the variation.

Answer 1

Let's assign a number from 1 to 5 to each automobile and let those ones with paint blemishes have numbers 1 and 2

List of all different samples of 3 automobiles out of 5:

1 2 3

1 2 4

1 2 5

1 3 4

1 3 5

1 4 5

2 3 4

2 3 5

2 4 5

3 4 5

We can see that there are one sample ({3, 4, 5}) with no blemished automobiles, 6 samples with one blemished automobile ({1, 3, 4}, {1, 3, 5}, {1, 4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5}) and 3 samples with 2 blemished automobiles ({1, 2, 3}, {1, 2, 4}, {1, 2, 5}). Which means that P(X = 0) = 1/10 = 0.1, P(X = 1) = 6/10 = 0.6, P(X=2) = 3/10 = 0.3

Mean:

$\mu=\sum_{i=0}^{2}P(X_i)X_i=\\ =0.1\cdot0+0.6\cdot1+0.3\cdot2=1.2$

Variance:

$\sigma^2=\sum_{i=0}^{2}P(X_i)(X_i-\mu)^2=\\ =0.1\cdot1.2^2+0.6\cdot0.2^2+0.3\cdot0.8^2=0.36$

Standard deviation:

$\sigma=\sqrt{\sigma^2}=0.6$

Coefficient of variation:

$CV=\frac{\sigma}{\mu}=0.5$