homework: section 4.2\nscore: 14/15 answered: 14/15\nquestion 15\nx p(x)\n0 0.3\n1 0.25\n2 0.1\n3 0.35\na…

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. $1.24$