Random Variables and Probability Distributions

This section first defines discrete and continuous random variables. Then, it introduces the distributions for discrete random variables. It also talks about the mean and variance calculations.

1.

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

3.

a. 0.4

b. 0.1

c. 0.9

d. 79.15

e. $σ2=1.5275$

f. $σ = 1.2359$

5.

a. 0.6528

b. 0.7153

c. $μ = 7.8333$

d. $σ2=5.4866$

e. $σ = 2.3424$

7.

a. 0.79

b. 0.60

c. $μ = 5.8, σ = 1.2570$

9.

\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.

a.

\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.

a.

\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.

a.

\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.

a.

\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.

a.

\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.