[Probability] Let N be a geometric random variable with parameter p. Given N,generate N many i.i.d. random numbers U1, U2, . . . , UN uniformly from [0,1]. Let M= max 1≤i≤N Ui. Find the cdf of M, i.e., find P(M≤x).
[Probability] Let N be a geometric random variable with parameter p. Given N,generate N many i.i.d....
Let X be the random variable with the geometric distribution with parameter 0 < p < 1. (1) For any integer n ≥ 0, find P(X > n). (2) Show that for any integers m ≥ 0 and n ≥ 0, P(X > n + m|X > m) = P(X > n) (This is called memoryless property since this conditional probability does not depend on m.)
Suppose that you need to generate a random variable Y with a density function f (y) corresponding to a beta distribution with range [0,1], and with a non-integer shape parameter for the beta distribution. For this case there is no closed-form cdf or inverse cdf. Suppose your choices for generating Y are either: a) an acceptance-rejection strategy with a constant majorizing function g(u) = V over [0, 1], i.e., generate u1 and u2 IID from a U[0,1] generator and accept...
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. (9 pts) Suppose the random variable Y has a geometric distribution with parameter p. Let ?? = √?? 3 3 . Find the probability distribution of V 3 4. (9 pts) Suppose the random variable Y has a geometric distribution with parameter p. Let V 3 Find the probability distribution of.
Problem 7. Let U1,U2,... be independent random variables all uniformly distributed on the unit interval, and let N be the first integer n 2 2 such that Un > Un-1. Show that for each real number 0<u < 1 !-un . 1- e-". (a) P(Ui-u and N = n) = (b) PUI S u and N is even) Problem 7. Let U1,U2,... be independent random variables all uniformly distributed on the unit interval, and let N be the first integer...
Problem 8 (10 points). Let X be the random variable with the geometric distribution with parameter 0 <p <1. (1) For any integer n > 0, find P(X >n). (2) Show that for any integers m > 0 and n > 0, P(X n + m X > m) = P(X>n) (This is called memoryless property since this conditional probability does not depend on m. Dobs inta T obabilita ndomlu abonn liaht bulb indofootin W
A discrete random variable X follows the geometric distribution with parameter p, written X ∼ Geom(p), if its distribution function is A discrete random variable X follows the geometric distribution with parameter p, written X Geom(p), if its distribution function is 1x(z) = p(1-P)"-1, ze(1, 2, 3, ). The Geometric distribution is used to model the number of flips needed before a coin with probability p of showing Heads actually shows Heads. a) Show that fx(x) is indeed a probability...
4.13 Let N be a geometric random variable with parameter p - 1/3. Calculate Pr[N s 2], PrN-2|, and PrN22
2. (20 pts.) Let X1,.., X45 be i.i.d. Uniform[0,1] random variables. Find (approximately) the probability P[12 X3++Xx18 2. (20 pts.) Let X1,.., X45 be i.i.d. Uniform[0,1] random variables. Find (approximately) the probability P[12 X3++Xx18
3. Let X be a random variable from a geometric distribution with parameter p (P(X- k p(1-P)"-, } k-1 k-1, 2, ...). Find Emin{X, 100