Question

homework: section 4.2
score: 14/15 answered: 14/15
question 15
x p(x)
0 0.3
1 0.25
2 0.1
3 0.35
a. find the expected value of the probability distribution. round to two decimal places.
b. find the standard deviation of the probability distribution. round to two decimal places.

Explanation:

Step1: Recall expected - value formula

The formula for the expected value $E(X)$ of a discrete probability distribution is $E(X)=\sum_{i}x_{i}P(x_{i})$.
\[

$$\begin{align*} E(X)&=(0\times0.3)+(1\times0.25)+(2\times0.1)+(3\times0.35)\\ &=0 + 0.25+0.2 + 1.05\\ &=1.50 \end{align*}$$

\]

Step2: Recall variance formula

The formula for the variance $\text{Var}(X)=\sum_{i}(x_{i}-E(X))^{2}P(x_{i})$.
First, calculate $(x_{i}-E(X))^{2}P(x_{i})$ for each $x_{i}$:

• For $x = 0$: $(0 - 1.5)^{2}\times0.3=( - 1.5)^{2}\times0.3 = 2.25\times0.3=0.675$
• For $x = 1$: $(1 - 1.5)^{2}\times0.25=( - 0.5)^{2}\times0.25 = 0.25\times0.25 = 0.0625$
• For $x = 2$: $(2 - 1.5)^{2}\times0.1=(0.5)^{2}\times0.1 = 0.25\times0.1=0.025$
• For $x = 3$: $(3 - 1.5)^{2}\times0.35=(1.5)^{2}\times0.35 = 2.25\times0.35 = 0.7875$

Then, $\text{Var}(X)=0.675 + 0.0625+0.025 + 0.7875=1.55$

Step3: Calculate standard - deviation

The standard deviation $\sigma=\sqrt{\text{Var}(X)}$.
$\sigma=\sqrt{1.55}\approx1.24$

Answer:

a. $1.50$
b. $1.24$