Random Variables and Probability Distributions

a. no: the sum of the probabilities exceeds 1

b. no: a negative probability

c. no: the sum of the probabilities is less than 1

a. 0.4

b. 0.1

c. 0.9

d. 79.15

e. $σ2=1.5275$

f. $σ = 1.2359$

a. 0.6528

b. 0.7153

c. $μ = 7.8333$

d. $σ2=5.4866$

e. $σ = 2.3424$

a. 0.79

b. 0.60

c. $μ = 5.8, σ = 1.2570$

$\begin{align*}\begin{array}{c|cccc}x & 0 & 1 & 2 & 3 \\\hline P(x) & 1 / 8 & 3 / 8 & 3 / 8 & 1 / 8\end{array}\end{align*}$

11.

$\begin{align*}\begin{array}{c|cccc}x & -1 & 999 & 499 & 99 \\\hline P(x) & \frac{4987}{5000} & \frac{1}{5000} & \frac{2}{5000} & \frac{10}{5000}\end{array}\end{align*}$

b. −0.4

c. 17.8785

13. 136

15.

$\begin{align*}\begin{array}{c|cc}x & C & C-150,000 \\\hline P(x) & 0.9825 & 0.0175\end{array}\end{align*}$

b. $C−2625$

c. $C ≥ 2625$

d. $C ≥ 2875$

17.

$\begin{align*}\begin{array}{c|cc}x & -1 & 1 \\\hline P(x) & \frac{20}{38} & \frac{18}{38}\end{array}\end{align*}$

b. $E(X)=−0.0526$ In many bets the bettor sustains an average loss of about 5.25 cents per bet.

c. 0.9986

19.

a. 43.54

b. 1.2046

21. 101.02

23.

$\begin{align*}\begin{array}{c|cccccc}x & 0 & 1 & 2 & 3 & 4 & 5 \\\hline P(x) & \frac{6}{36} & \frac{10}{36} & \frac{8}{36} & \frac{6}{36} & \frac{4}{36} & \frac{2}{36}\end{array}\end{align*}$

b. 1.9444

c. 1.4326

25.

$\begin{align*}\begin{array}{c|ccc}x & 0 & 1 & 2 \\\hline P(x) & 0.902 & 0.096 & 0.002\end{array}\end{align*}$

b. 0.902

27.

a. 2523.25

b. 227,092.5

c. 270,000

d. The owner will install the cover.