Question

consider the discrete random variable x given in the table below. calculate the mean, variance, and standard deviation of x. round answers to two decimal places.
what is the expected value of x?

Explanation:

Step1: Calculate the mean ($\mu$ or expected value $E(X)$)

The formula for the mean of a discrete - random variable is $E(X)=\sum_{i}x_ip_i$.
\[

$$\begin{align*} E(X)&=(1\times0.13)+(3\times0.32)+(5\times0.1)+(6\times0.13)+(7\times0.14)+(8\times0.08)+(16\times0.1)\\ &=0.13 + 0.96+0.5 + 0.78+0.98+0.64+1.6\\ &=5.69 \end{align*}$$

\]

Step2: Calculate the variance ($\sigma^{2}$)

The formula for the variance of a discrete - random variable is $\sigma^{2}=\sum_{i}(x_i - \mu)^2p_i$.
\[

$$\begin{align*} &(1 - 5.69)^2\times0.13+(3 - 5.69)^2\times0.32+(5 - 5.69)^2\times0.1+(6 - 5.69)^2\times0.13+(7 - 5.69)^2\times0.14+(8 - 5.69)^2\times0.08+(16 - 5.69)^2\times0.1\\ &=(-4.69)^2\times0.13+(-2.69)^2\times0.32+(-0.69)^2\times0.1+(0.31)^2\times0.13+(1.31)^2\times0.14+(2.31)^2\times0.08+(10.31)^2\times0.1\\ &=(21.9961\times0.13)+(7.2361\times0.32)+(0.4761\times0.1)+(0.0961\times0.13)+(1.7161\times0.14)+(5.3361\times0.08)+(106.2961\times0.1)\\ &=2.859493+2.315552 + 0.04761+0.012493+0.240254+0.426888+10.62961\\ &=16.52190 \end{align*}$$

\]

Step3: Calculate the standard deviation ($\sigma$)

The standard deviation is the square - root of the variance, $\sigma=\sqrt{\sigma^{2}}$.
\[
\sigma=\sqrt{16.52190}\approx4.06
\]

Answer:

$\mu = 5.69$
$\sigma^{2}=16.52$
$\sigma = 4.06$
Expected value of $X$: $5.69$