Question

1. trivia quiz the probabilities that a player will get 5 to 10 questions right on a trivia quiz are shown below. find the mean, variance, and standard deviation for the distribution.

x | 5 6 7 8 9 10
p(x) | 0.05 0.2 0.4 0.1 0.15 0.1

Explanation:

Step1: Calculate the mean ($\mu$)

The formula for the mean of a discrete - probability distribution is $\mu=\sum xP(x)$.
\[

$$\begin{align*} \mu&=(5\times0.05)+(6\times0.2)+(7\times0.4)+(8\times0.1)+(9\times0.15)+(10\times0.1)\\ &=0.25 + 1.2+2.8 + 0.8+1.35 + 1\\ &=7.4 \end{align*}$$

\]

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

The formula for the variance is $\sigma^{2}=\sum(x - \mu)^{2}P(x)$.
\[

$$\begin{align*} (5 - 7.4)^{2}\times0.05&=(- 2.4)^{2}\times0.05=5.76\times0.05 = 0.288\\ (6 - 7.4)^{2}\times0.2&=(-1.4)^{2}\times0.2 = 1.96\times0.2=0.392\\ (7 - 7.4)^{2}\times0.4&=(-0.4)^{2}\times0.4 = 0.16\times0.4 = 0.064\\ (8 - 7.4)^{2}\times0.1&=(0.6)^{2}\times0.1=0.36\times0.1 = 0.036\\ (9 - 7.4)^{2}\times0.15&=(1.6)^{2}\times0.15 = 2.56\times0.15=0.384\\ (10 - 7.4)^{2}\times0.1&=(2.6)^{2}\times0.1 = 6.76\times0.1=0.676 \end{align*}$$

\]
\[

$$\begin{align*} \sigma^{2}&=0.288+0.392 + 0.064+0.036+0.384+0.676\\ &=1.84 \end{align*}$$

\]

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

The standard deviation is the square - root of the variance, $\sigma=\sqrt{\sigma^{2}}$.
\[
\sigma=\sqrt{1.84}\approx1.3565
\]

Answer:

Mean: $7.4$, Variance: $1.84$, Standard Deviation: approximately $1.3565$