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) = 1/3?

To determine

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

Answer to Problem 1CQQ

No, the random variable x does not follow the probability distribution.

Explanation of Solution

Given info:

The random variable x takes values of 0, 1, 2, and 3. Also, each of the value for the random variable has an equal probability of $\frac{1}{3}$.

Requirements:

The following requirements should be satisfied for the distribution to follow the 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 the 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 from 0 to 3. Also, each value of x has its corresponding probability. Hence, the requirement 1 is satisfied.

The sum of all probabilities is:

$\begin{array}{c}{\displaystyle \sum P\left(x\right)}=\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}\\ =\frac{4}{3}\\ =1.3333\end{array}$

Hence, the requirement 2 is not satisfied.

In the given distribution, all the values of P(x) for the random variable x lie between 0 and 1. Hence, the requirement 3 is satisfied.

Since, the requirement 2 is not satisfied the random variable x does not follow the probability distribution.