in this problem, round to four decimal places. consider the discrete random variable x given in the table…

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.\n| x | 7 | 16 | 19 | 20 |\n| p(x) | 0.66 | 0.1 | 0.13 | 0.11 |\nμ = \nσ² = \nσ =

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.\n| x | 7 | 16 | 19 | 20 |\n| p(x) | 0.66 | 0.1 | 0.13 | 0.11 |\nμ = \nσ² = \nσ =

Answer

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:

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