I roll two fair six-sided dice, and let X be defined as the sum of the two outcomes I observe. Find the range of X, as well as its probability mass function p(x). Also, show that it is a valid pmf.

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X1 and X2 are independent random variables with X1 ~ Exp(A1) and X2 ~ Exp(A2). Let M = min{X1, X2}. Determine the probability density function of M.

Pb (Z = z) = B (8.25z - z2) with z E {1, 2, ......, 5}.   Find the mean and variance of Z using the probability mass function equation above.

Let Y have a Beta distribution with parameters a and B. Let U = 1 - Y. Compute fu(u), the probability density function of U. You must completely define the function on its domain. What distribution (with what parameters) does U have?

Three couples and two single individuals have been invited to an investment seminar and have agreed to attend. Suppose the probability that any particular couple or individual arrives late is 0.32 (a couple will travel together in the same vehicle, so either both people will be on time or else both will arrive late). Assume that different couples and individuals are on time or late independently of one another. Let X the number of people who arrive late for the seminar. (a) Determine the probability mass function of X. [Hint: label the three couples #1, #2, and #3 and the two individuals #4 and #5.] (Round your answers to four decimal places.) Р(Х %3D х) 3 4 6. (b) Obtain the cumulative distribution function of X. (Round your answers to four decimal places.) F(x) 3 4 8. Use the cumulative distribution function of X to calculate P(2 < X < 6). (Round your answer to four decimal places.) P(2 < X < 6) =

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. -