If $X$ is a random variable with distribution given below Then the value of $k$ and its variance are…
If $X$ is a random variable with distribution given below
Then the value of $k$ and its variance are respectively given by
$\frac{1}{8}, \frac{22}{27}$
$\frac{1}{8}, \frac{23}{27}$
$\frac{1}{8}, \frac{8}{9}$
$\frac{1}{8}, \frac{3}{4}$
Solution
The sum of all the probabilities in a probability distribution is always unity.
$\begin{array}{ll}
\therefore \quad & k+3 k+3 k+k=1 \\
& \Rightarrow 8 k=1 \\
& \Rightarrow k=\frac{1}{8}
\end{array}$
$\begin{aligned} \mathrm{E}(\mathrm{X}) & =\sum x_{\mathrm{i}} \cdot \mathrm{P}\left(x_{\mathrm{i}}\right) \\ & =0\left(\frac{1}{8}\right)+1\left(\frac{3}{8}\right)+2\left(\frac{3}{8}\right)+3\left(\frac{1}{8}\right)=\frac{3}{2}\end{aligned}$
$\operatorname{Var}(X)=E\left(X^2\right)-[E(X)]^2$
$\begin{aligned} & =0^2\left(\frac{1}{8}\right)+1^2\left(\frac{3}{8}\right)+2^2\left(\frac{3}{8}\right)+3^2\left(\frac{1}{8}\right)-\left(\frac{3}{2}\right)^2 \\ & =\frac{3}{4}\end{aligned}$
