Consider the joint probability distribution below. Complete parts (a) through (c). Y Y 0 1 0 1 Pw 0.30 0.40 a. Compute the marginal probability distributions for X and Y. 0.30 0.40 X X 2 0.10 0.20 2 0.10 0.20 P(y) P(x) (Type integers or decimals.) b. Compute the covariance and correlation for X and Y. Cov(X,Y)= (Round to four decimal places as needed.) Corr(X,Y)= (Round to three decimal places as needed.) c. Compute the mean and variance for the linear function W=X+Y. (Round to two decimal places as needed.) (Round to four decimal places as needed.)

Let μx and of be the expectation and variance of a forecast model F for an observations y, and let pÃ(.) denote the probability density function for the forecast model. Also let G be a different forecast model. Which of the following statements are true? a. Two forecast scores S(F, y) and S(G, y) are independent. b. The square error (y – µµ)² is a proper score. C. The score (y - μF) ²/0 is proper score that improves in the squared error score by taking the variance into account. d. The logarithmic score – log[pf(y)] is a strictly proper score. -

The joint probability distribution of the number X of cars and the number Y of buses per signal cycle at a proposed left-turn lane is displayed in the accompanying joint probability table. P(x, y) 1 2 0.015 0.010 0.025 1 0.030 0.020 0.050 2 0.075 0.050 0.125 3 0.090 0.060 0.150 0.060 0.040 0.100 0.030 0.020 0.050 What is the probability that there is exactly one car and exactly one bus during a cycle? What is the probability that there is at most one car and at most one bus during a cycle? What is the probability that there is exactly one car during a cycle? Exactly one bus? P(exactly one car) = P(exactly one bus) = - Suppose the left-turn lane is to have a capacity of five cars and one bus is equivalent to three cars. What is the probability of an overflov during a cycle?

The probability density function of a discrete random variable X is given by the following table: Px(X = 1) = .05 Px(X = 2) = .10 Px(X = 3) = .12 Px(X = 4) = .30 Px (X = 5) = .30 Px (X = 6) = .1i Px (X = 7) = .01 Px(X = 8) = .01 i) Compute E(X). ii) Compute Var(X). iii) Compute Px(X 3)

The probability distribution for X and Y is shown by the table. Find the marginal distribution function value of X = 1. G(1) = X 2 3 1 0.06 0.10 0.04 2 0.10 0.12 0.13 3 0.15 0.19 0.11 Y

Find the marginal probabilities of each random variable X and Y in thefollowing table:- Compute the covariance and the correlation of these two random variables.

Q2

Let X be a random variable with the following probability distribution. Value x of X P(X=x) 2 0.05 3 0.05 4 0.05 5 0.85 Complete the following. (If necessary, consult a list of formulas.) (a) Find the expectation E(X) of X. E (X) = 0 (b) Find the variance Var(X) of X. Var(x) = ☐ Check B X G

Q1. Suppose X is a continuous random variable. Find an example of a probability density function for X giving expected value E(X) = 1 and variance V (X) = 3 if X has . . .  (a.) a uniform distribution. (b.) an exponential distribution. (c.) a normal distribution. In each case, if there is no such probability density function, explain why this is so.

A manufacturer of phone batteries determines that the average length of talk time for one of its batteries is 470 minutes.  Suppose that the standard deviation is known to be 32 minutes and that the data are approximately bell-shaped. Estimate the percentage of batteries whose talk time is between 406 minutes and 534 minutes. Estimate the percentage of batteries whose talk time is less than 438 minutes. Estimate the percentage of batteries whose talk time is more than 534 minutes. answer in excel