Random Variables and Probability Distributions
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. \(σ^2=5.4866\)
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.
b. 0.902
27.
a. 2523.25
b. 227,092.5
c. 270,000
d. The owner will install the cover.