Chapter 10, Problem 1RE

Solution

To explain that given sample space is a valid probability function.

The given sample is not a valid probability function.

Given:

The sample space for a random variable $X$ is given:

  $\left\{1,2,5,10\right\}$

Concept Used:

For a valid Probability distribution, following conditions should be fulfilled:

• Probabilities must have range between 0 and 1
• Sum of all the provided probabilities must be equal to 1

Calculation:

Draw a table for the probability of the given random variables:

  $\begin{array}{cc}\text{Random Variable}& \text{Probability}\\ 1& 0.45\\ 2& 0.25\\ 5& 0.15\\ 10& 0.05\end{array}$

Now, add all the probabilities as:

  $\begin{array}{l}P\left(E\right)=P\left(1\right)+P\left(2\right)+P\left(5\right)+P\left(10\right)\\ P\left(E\right)=0.45+0.25+0.15+0.05\\ P\left(E\right)=0.90\end{array}$

Here, the sum of the probabilities of all outcomes is not $1$ , hence the given sample space is not a valid probability function.