Question

in this problem, round to four decimal places. consider the discrete random variable x given in the table below. calculate the mean, variance, and standard deviation of x. what is the expected value of x? enter an integer or decimal number more...

Explanation:

Step1: Calculate the mean $\mu$

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

$$\begin{align*} \mu&=(7\times0.66)+(16\times0.1)+(19\times0.13)+(20\times0.11)\\ & = 4.62+1.6 + 2.47+2.2\\ &=10.89 \end{align*}$$

\]

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

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

$$\begin{align*} (7 - 10.89)^2\times0.66&=(- 3.89)^2\times0.66 = 15.1321\times0.66=9.9872\\ (16 - 10.89)^2\times0.1&=(5.11)^2\times0.1 = 26.1121\times0.1 = 2.6112\\ (19 - 10.89)^2\times0.13&=(8.11)^2\times0.13=65.7721\times0.13 = 8.5504\\ (20 - 10.89)^2\times0.11&=(9.11)^2\times0.11 = 82.9921\times0.11=9.1291\\ \sigma^{2}&=9.9872 + 2.6112+8.5504 + 9.1291\\ &=30.2779 \end{align*}$$

\]

Step3: Calculate the standard - deviation $\sigma$

The standard deviation $\sigma=\sqrt{\sigma^{2}}$.
\[
\sigma=\sqrt{30.2779}\approx5.5025
\]

Answer:

$\mu = 10.8900$
$\sigma^{2}=30.2779$
$\sigma = 5.5025$
The expected value of $X$ is $10.8900$