Question

A random variable $X$ has the following probability distribution.

$X$	$1$	$2$	$3$	$4$
$P\left(X\right)$	$k$	$2k$	$2k$	$4k$

Then the mean of $X$ is

• A. $3$
• B. $1$
• C. $4$
• D. $2$

Answer 1

The correct option is A

$3$

Explanation for the correct option:

Step 1: Find the value of $k$:

Obtain the value of $k$ by using the probability relation.

$$\begin{gathered} \sum _{x=1}^{4}P\left(x\right)=1 \\ \Rightarrow k+2k+2k+4k=1 \\ \Rightarrow 9k=1 \\ \Rightarrow k=\frac{1}{9} \end{gathered}$$

Step 2: Find the mean using the binomial probability method:

Multiply the value of $x$ with $p\left(x\right)$ and sum up to find the mean.

$\begin{array}{rcl}\mu & =& 1\times \frac{1}{9}+2\times \frac{2}{9}+3\times \frac{2}{9}+4\times \frac{4}{9}\\ & =& \frac{27}{9}\\ & =& 3\end{array}$

Hence, option (A) is the correct answer.