Question

Statistic problems an explanation would be appreciated

(5pts) Let \(X \sim N(25,3)\). Find the expected value, variance, and standard deviation of

\(X .\) Find \(P(X \leq 26.5)\)

(5pts) Let \(X \sim N(12,2)\). Find the expected value, variance, and standard deviation of \(X\). Find \(P(X>12.8)\)

(5pts) Let \(Y \sim\) binomial \((15, .65)\). Find the expected value, variance, and standard deviation of \(Y\). Find \(P(Y=10)\).

Answer 1

Q1)

\(E(X)=\mu=25\)

\(V(X)=\sigma^{2}=9\)

\(S D(X)=\sigma=3\)

\(P(X \leq 26.5)=P\left(Z \leq \frac{26.5-25}{3}\right)\)

\(P(Z \leq 0.5)=P(-\infty

\(P(Z \leq 0.5)=0.5+0.1915=0.6915\)

Q 2)

\(E(X)=\mu=12\)

\(V(X)=\sigma^{2}=4\)

\(S D(X)=\sigma=2\)

\(P(X>12.8)=P\left(Z>\frac{12.8-12}{2}\right)\)

\(P(Z>0.4)=P(0

\(P(Z>0.4)=0.5-0.1554=0.3446\)

Q 3)

\(E(Y)=n p=15 * 0.65=9.75\)

\(V(Y)=n p q=15^{*} 0.65^{*} 0.35=3.4125\)

\(S D(Y)=\sqrt{3.4125}=1.8473\)

\(P(Y=10)=\left(\begin{array}{c}15 \\ 10\end{array}\right)(0.65)^{10}(0.35)^{15-10}=0.2123\)

Excel Command : =BINOM.DIST(10,15,0.65,FALSE)