Question

consider the discrete random variable x given in the table below. calculate the mean, variance, and standard deviation of x. round to 4 decimals.

x	2	4	7	18
p(x)	0.11	0.07	0.12	0.7

μ =
σ² =
σ =

what is the expected value of x?

Explanation:

Step1: Calculate the mean (expected value)

The formula for the mean $\mu = E(X)=\sum_{i}x_{i}P(x_{i})$.
\[

$$\begin{align*} \mu&=(2\times0.11)+(4\times0.07)+(7\times0.12)+(18\times0.7)\\ &=0.22 + 0.28+0.84 + 12.6\\ &=13.94 \end{align*}$$

\]

Step2: Calculate the variance

The formula for the variance $\sigma^{2}=\sum_{i}(x_{i}-\mu)^{2}P(x_{i})$.
\[

$$\begin{align*} &(2 - 13.94)^{2}\times0.11+(4 - 13.94)^{2}\times0.07+(7 - 13.94)^{2}\times0.12+(18 - 13.94)^{2}\times0.7\\ &=(- 11.94)^{2}\times0.11+(-9.94)^{2}\times0.07+(-6.94)^{2}\times0.12+(4.06)^{2}\times0.7\\ &=(142.5636)\times0.11+(98.8036)\times0.07+(48.1636)\times0.12+(16.4836)\times0.7\\ &=15.6820+6.9163+5.7796 + 11.5385\\ &=39.9164 \end{align*}$$

\]

Step3: Calculate the standard deviation

The standard deviation $\sigma=\sqrt{\sigma^{2}}$.
$\sigma=\sqrt{39.9164}\approx6.3179$

Answer:

$\mu = 13.9400$
$\sigma^{2}=39.9164$
$\sigma = 6.3179$
The expected value of $X$ is $13.9400$