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.

x	7	16	19	20
p(x)	0.66	0.1	0.13	0.11

μ =
σ² =
σ =

Explanation:

Step1: Calculate the mean

The formula for the mean $\mu$ of a discrete - random variable is $\mu=\sum_{i}x_ip_i$.
$\mu = 7\times0.66+16\times0.1 + 19\times0.13+20\times0.11$
$=4.62 + 1.6+2.47 + 2.2$
$=10.8900$

Step2: Calculate the variance

The formula for the variance $\sigma^{2}$ is $\sigma^{2}=\sum_{i}(x_i-\mu)^{2}p_i$.
$(7 - 10.89)^{2}\times0.66+(16 - 10.89)^{2}\times0.1+(19 - 10.89)^{2}\times0.13+(20 - 10.89)^{2}\times0.11$
$=(- 3.89)^{2}\times0.66+(5.11)^{2}\times0.1+(8.11)^{2}\times0.13+(9.11)^{2}\times0.11$
$=15.1321\times0.66 + 26.1121\times0.1+65.7721\times0.13+82.9921\times0.11$
$=9.9872+2.6112 + 8.5504+9.1291$
$=30.2779$

Step3: Calculate the standard deviation

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

Answer:

$\mu = 10.8900$
$\sigma^{2}=30.2779$
$\sigma = 5.5025$