ISEN 350 Problem Set M6

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)

(7) The median of the random variable X whose probability density function (3 3 = { 2x - ²x²,0 < x < 2², 0 , 0. W. is: (a) 0.5 f(y) = (b) 1 (c) 2 (d) 1.5

If F(x) is cumulative distrbution function of a continuous random variable X, the probability (function f(x) is the dervative of F(x (Fa)se (Trụe

If E denotes the expectation the variance of a random variable X is denoted as? a. (E(X))² b. 2E(X) C. E(X²) O c. d. E(X²)-(E(X))² <

The probability density function of a discrete random variable X is given by the following table: Px(X = 0) = .54 Px(X = 1) = .16 Px(X = 2) = .06 Px(X = 3) = .04 Px(X = 4) = .20 %3D i) Compute E(X). ii) Compute Var(X).

If P(x) = 0.5 and x = 4, then the expected value of x is:   a.E(x) = 4.5   b.E(x) = 0.5   c.E(x) = 4   d.E(x) = 0   e.E(x) = 2

(1) Which of the following functions is a valid probability density function (p.d.f.)? f(x), f(x) 1 0.5 -1 (a) 1 2 (b) -1 (c) f(x) ==, 0 2 (a) ; (b)} (c) (d) 3 (4) The following table represents the price distribution of all different bags in a store. What are the population and population size, respectively? (a) A bag, 65 Price (S) Frequency 25 15 – 19.99 (b) All bags in the store, 65 20 – 24.99 19 (c) All bags in the store, not known 25 – 29.99 13 30 – 34.99 8 (d) A bag, not known (5) 9 students are waiting in a line to get on the bus. Among them, 3 are close friends and insist on sticking together in the line. In how many ways can this line be formed? () C) (b) )-3! |×6! (с) 3!x6! (d) 3!×7! NIM

2. A discrete random variable X has probability density function of the form (c(8 – x) for x = 0, 1, 2, 3, 4, 5 f(x) = otherwise. (a) Find the constant c. (b) Find P(X > 2). (c) Find the expected value E(X) for the random variable X.

If the random variable Y has the following probability density function f(y) = {5 ,0 < y< 5 , (0,otherwise The probability that the root of the function g(x) = 4x² + 4yx + (y + 2) are real is

7. Consider the two-class logistic model with the linear predictor: p(x) log (₁ - p(x)) = ³0 + B₁X1 + B₂X2 + ... + BpXp· 1 (a). Write down an expression for the odds O(x): = p(1|x) p(2x) (b). Show that the odds ratio comparing the odds for two new cases with explanatory O(x') variables x and x' does not depend on Bo. (c). What does this imply if we simply want to rank new cases in terms of p(1|x).

The probability function for the number of insurance policies John will sell to a customer is given by f(x) = 0.5 for X = 6. 0, 1, or 2. (a) Is this a valid probability function? Explain your answer. Yes, f(x) 2 0 and Ef(x) + 1 Yes, f(x) > 0 and Ef(x) = 1 No, f(x) > 0 and Ef(x) # 1 No, f(x) > 0 and Ef(x) = 1 (b) What is the probability that John will sell exactly 2 policies to a customer? (Round your answer to three decimal places.) (c) What is the probability that John will sell at least 2 policies to a customer? (Round your answer to three decimal places.) (d) What is the expected number of policies John will sell? (Round your answer to three decimal places.) (e) What is the variance of the number of policies John will sell? (Round your answer to three decimal places.)

Suppose that the distribution of the lifespan of an item has the hazard rate function λ(t) = t3.   What is the probability that: (a) the item survives to age 2? (b) a 1-year-old item will survive to age 2?

(a) What are the values of P(D), P(T), and P(Dn T)? P(D) = 0.168 P(T) = 0.394 P(DnT) = 0.076 (b) Use the given probability information to set up a hypothetical 1,000 table with columns corresponding to D and DC and rows corresponding to T and TC. Change in Diagnosis (D) No Change in Diagnosis (DC) Change in Treatment (7) No Change in Treatment (7) Total X X Total X 1,000

Question 23. (a) Use the expectation operator to show that for any random variable Y, Var(Y) = E(Y²) – [E(Y)]². Your proof must start with the definition of variance.