a random variable Z with non-zero Consider probability on all the positive integers, that is PIZ...
4.8. Let Z be a random variable with the geometric probability mass function where 0 < π < 1. (a) Show that Z has a constant failure rate in the sense that PriZ kZk1 T for k 0, 1,.... (b) Suppose Z' is a discrete random variable whose possible values are 0, 1, and for which Pr(Z'=KZ2k} = 1-π for k 0,1,.... Show that the probability mass function for Z' is p(k).
(10 points) Consider a discrete random variable X, which can only take on non-negative integer values, with E[Xk] = 0.8, k = 1,2, .... Use the moment generating function approach to find the pmf of Px(k), k = 0,1,....
Consider the random variable Y, whose probability density function is defined as: if 0 y1 2 y if 1 y < 2 fr(v) 0 otherwise (a) Determine the moment generating function of Y (b) Suppose the random variables X each have a continuous uniform distribution on [0,1 for i 1,2. Show that the random variable Z X1X2 has the same distribution = as the random variable Y defined above. Consider the random variable Y, whose probability density function is defined...
4. Let X be a discrete random variable with PrX-ici for positive, odd integers 3 Sis 13; otherwise, the probability is zero a) Compute the value of c. (b) What is the mean of X? (c) What is the second moment of X (that is, E[X21)? (d) What is the variance of X? (e) Compute E[min(9, X) (f) Compute E(X-9)*) where x+ = max(x,0).
Question 5. Let r, n be positive integers. 1. (6 pts) Consider the random binary r n matrix M , where each entry is equal to 0 or 1 with probability 1/2 (so each entry follows the Bernoulli random variable with parameter 1/2), and these entries are (jointly) independent random variables. What is the probability that each column in M has at most one entry with 1? 1 2. (*, 4 pts) Let S1, . . . , Srbe identically...
Let X be a discrete random variable with Pr{X = i} = ci for positive, odd integers 3 ≤ i ≤ 13; otherwise, the probability is zero. (a) Compute E[min(9,X)]. (b) Compute E[(X − 9)+] where x+ = max(x, 0).
Question 1: 1a) Let the random variable X have a geometric distribution with parameter p , i.e., P(X = x) = pq??, x=1,2,... i) Show that P(X > m)=q" , where m is a positive integer. (5 points) ii) Show that P(X > m+n X > m) = P(X>n), where m and n are positive integers. (5 points) 1b) Suppose the random variable X takes non-negative integer values, i.e., X is a count random variable. Prove that (6 points) E(X)=...
4. Let X be a discrete random variable with PrXici for positive, odd integers 3 3i s 13 otherwise, the probability is zero (a) Compute the value of c. (b) What is the mean of X? (c) What is the second moment of X (that is, E[X2])? (d) What is the variance of X? (e) Compute E[min(9, X)]. (f) Compute E[(X - 9)+] wherea+max(x, 0)
4. Let X be a discrete random variable with Pr(X i} = ci for positive, odd integers 3 i 13; otherwise, the probability is zero e) Compute E[min(9, X)]. (f) Compute E(X-9)+] where x+ = max (x,0).
. 3. For each positive integer k, consider Xt as a continuous random variable with PDF given by focsi=fksk-! 065:-1, . kolo, otherwise. Let W denote the number of k=1,2,..., 12 for which X = 23. Calculate <WY ..