Question

bell ringer - sep 23
find k, and find the mean, variance, and standard deviation of the following.

1. x -3 5 4 2 7

p(x) 0.2 0.3 k 0.25 0.15

1. x 0 1 2 -1

p(x) 0.1 0.24 0.36 k

Explanation:

Step1: Find k in the first distribution

Since the sum of all probabilities in a probability distribution is 1, we have $0.2 + 0.3 + k+0.25 + 0.15=1$.
$k = 1-(0.2 + 0.3+0.25 + 0.15)=1 - 0.9 = 0.1$.

Step2: Calculate the mean $\mu_1$ of the first distribution

The mean $\mu=\sum_{i}x_ip_i$. So, $\mu_1=(-3)\times0.2 + 5\times0.3+4\times0.1 + 2\times0.25+7\times0.15$.
$\mu_1=- 0.6+1.5 + 0.4+0.5 + 1.05=2.9$.

Step3: Calculate the variance $\sigma_1^{2}$ of the first distribution

The variance $\sigma^{2}=\sum_{i}(x_i-\mu)^2p_i$.
$( - 3 - 2.9)^2\times0.2+(5 - 2.9)^2\times0.3+(4 - 2.9)^2\times0.1+(2 - 2.9)^2\times0.25+(7 - 2.9)^2\times0.15$.
$(-5.9)^2\times0.2+(2.1)^2\times0.3+(1.1)^2\times0.1+( - 0.9)^2\times0.25+(4.1)^2\times0.15$.
$34.81\times0.2 + 4.41\times0.3+1.21\times0.1 + 0.81\times0.25+16.81\times0.15$.
$6.962+1.323 + 0.121+0.2025+2.5215 = 11.13$.

Step4: Calculate the standard - deviation $\sigma_1$ of the first distribution

The standard deviation $\sigma=\sqrt{\sigma^{2}}$, so $\sigma_1=\sqrt{11.13}\approx3.34$.

Step5: Find k in the second distribution

Since the sum of all probabilities is 1, $0.1+0.24 + 0.36 + k = 1$.
$k=1-(0.1 + 0.24+0.36)=1 - 0.7 = 0.3$.

Step6: Calculate the mean $\mu_2$ of the second distribution

$\mu_2=0\times0.1+1\times0.24 + 2\times0.36+( - 1)\times0.3$.
$\mu_2=0 + 0.24+0.72-0.3 = 0.66$.

Step7: Calculate the variance $\sigma_2^{2}$ of the second distribution

$\sigma_2^{2}=(0 - 0.66)^2\times0.1+(1 - 0.66)^2\times0.24+(2 - 0.66)^2\times0.36+( - 1 - 0.66)^2\times0.3$.
$( - 0.66)^2\times0.1+(0.34)^2\times0.24+(1.34)^2\times0.36+( - 1.66)^2\times0.3$.
$0.4356\times0.1+0.1156\times0.24 + 1.7956\times0.36+2.7556\times0.3$.
$0.04356+0.027744+0.646416+0.82668 = 1.5444$.

Step8: Calculate the standard - deviation $\sigma_2$ of the second distribution

$\sigma_2=\sqrt{1.5444}\approx1.24$.

Answer:

1. $k = 0.1$, mean $\mu_1 = 2.9$, variance $\sigma_1^{2}=11.13$, standard - deviation $\sigma_1\approx3.34$.
2. $k = 0.3$, mean $\mu_2 = 0.66$, variance $\sigma_2^{2}=1.5444$, standard - deviation $\sigma_2\approx1.24$.