Chapter 5, Problem 1CQQ

Is a probability distribution defined if the only possible values of a random variable are 0, 1, 2, 3, and P(0) = P(1) = P(2) = P(3) = 0.25?

To determine

Whether or not the random variable x is a probability distribution.

Answer to Problem 1CQQ

Yes, the random variable x follows the probability distribution.

Explanation of Solution

Given info:

The random variable x takes values 0, 1, 2, and 3. Also, each of the value of the random variable has equal probability of 0.25.

Requirements:

The following requirements should be satisfied for the distribution to follow probability distribution.

1. 1. The given random variable (x) must take up numerical values and it should have its corresponding probabilities.
2. 2. The sum of all probabilities must be equal to 1. That is, ${\displaystyle \sum P\left(x\right)}=1$
3. 3. The probability values must lie between 0 and 1, inclusive. That is, $0\le P\left(x\right)\le 1$

Here, the random variable x takes the numerical values, 0 to 3. Also each value of x has its corresponding probability. Hence, the requirement 1 is satisfied.

The sum of all probabilities is,

$\begin{array}{c}{\displaystyle \sum P\left(x\right)}=0.25+0.25+0.25+0.25\\ =1\end{array}$

Here, it is observed random variable x satisfies all the requirement of probability distribution function.

Thus, the random variable x follows the probability distribution