Question

Marketing estimates that a new instrument for the analysis of soil samples will be very successful, moderately successful, or unsuccessful, which is probabilities 0.3, 0.6, and 0.1, respectively. The yearly revenue associated with a very successful, moderately successful, or unsuccessful is $10 million, $5 million, and $1 million, respectively. Let the random variable X denote the yearly revenue of the product. Determine:

a) the mass function of X;

b) the mean of X; and

c) the variance of X.

Answer 1

Solution

Let X denote the yearly revenue of the product.

(a) The mass function of X

The probability mass function of X is as shown in the diagram below.

(b) The mean of X

Mean $\mu=\sum XP(X)$

$\mu=10(0.3)+5(0.6)+1(0.1)$

$\mu=6.1$

(c) The variance of X

$\sigma^2=\sum X^2.P(X)-\mu$

$\sigma^2=[10^2(0.3)+5^2(0.6)+1^2(0.1)]-6.1^2$

$\sigma^2=45.1-37.21$

$\sigma^2=7.89$