Chapter 4, Problem 1P

Let X be the random variable representing the number of hypertensive adults in Example 3.12.

Derive the probability-mass function for X.

To determine

Delineate the probability mass function of X .

Answer to Problem 1P

The probability mass function of X is given as follows:

x	$P_r\left(X=x\right)$
0	0.72
1	0.26
2	0.02

Explanation of Solution

Let X be the random variable representing the number of adults.

Consider that Events A and B represent the male and female.

It is given that the probability of male adult has hypertensive as 0.1 and the probability of female adult as hypertensive as 0.2.

Here, X can take 0, 1, and 2, “0” represents that both male and female are not affected, “1” represents one adult is affected and one adult is not affected, and “2” represents that both male and female are affected.

When X=0, the probability mass function is as follows:

$\begin{array}{c}{P}_r\left(X=0\right)=P_r\left(\begin{array}{l}\text{Both male and female }\\ \text{adults are not affected}\end{array}\right)\\ =\left(1-0.1\right)+\left(1-0.2\right)\\ =0.9\times 0.8\\ =0.72\end{array}$

When X=1, the probability mass function is as follows:

$\begin{array}{c}{P}_r\left(X=1\right)=P_r\left(\text{at least one adult is affected}\right)\\ =0.1\left(1-0.2\right)+0.2\left(1-0.1\right)\\ =0.08+0.18\\ =0.26\end{array}$

When X=2, the probability mass function is as follows:

$\begin{array}{c}{P}_r\left(X=2\right)=P_r\left(\text{Both adults are affected}\right)\\ =0.1\times 0.2\\ =0.02\end{array}$

Therefore, the probability mass function is given as follows:

x	$P_r\left(X=x\right)$
0	0.72
1	0.26
2	0.02