consider the discrete random variable x given in the table below. calculate the mean, variance, and standard…

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.\n\n| x | 1 | 3 | 5 | 6 | 7 | 8 | 16 |\n| p(x) | 0.13 | 0.32 | 0.1 | 0.13 | 0.14 | 0.08 | 0.1 |\n\nμ = \nσ² = \nσ = \n\nwhat is the expected value of x?

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.\n\n| x | 1 | 3 | 5 | 6 | 7 | 8 | 16 |\n| p(x) | 0.13 | 0.32 | 0.1 | 0.13 | 0.14 | 0.08 | 0.1 |\n\nμ = \nσ² = \nσ = \n\nwhat is the expected value of x?

Answer

Explanation:

Step1: Calculate the mean (expected value)

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

The formula for the variance $\sigma^{2}=\sum_{i}(x_{i}-\mu)^{2}P(X = x_{i})$. [ \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\ &\approx16.52 \end{align*} ]

Step3: Calculate the standard - deviation

The standard deviation $\sigma=\sqrt{\sigma^{2}}$. Since $\sigma^{2}\approx16.52$, then $\sigma=\sqrt{16.52}\approx4.06$.

Answer:

Mean $\mu = 5.69$, Variance $\sigma^{2}=16.52$, Standard Deviation $\sigma = 4.06$