Problem 8 (10 points). Let X be the random variable with the geometric distribution with parameter...
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.)
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)=...
Need help with this Problem 4 A discrete random variable X follows the geometric distribution with parameter p, written X ~Geom(p), if its distribution function is fx(x) = p(1-p)"-1, xe(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 Ix(z) is indeed a probability inass function, i.e., the sum over all possible values of z is one...
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...
3. Let X be a geometric random variable with parameter p. Prove that P(X >k+r|X > k) = P(X > r). This is called the memoryless property of the geometric random variable.
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
Problem 3 [5 points) (a) [2 points] Let X be an exponential random variable with parameter 1 =1. find the conditional probability P{X>3|X>1). (b) [3 points] Given unit Gaussian CDF (x). For Gaussian random variable Y - N(u,02), write down its Probability Density Function (PDF) [1 point], and express P{Y>u+30} in terms of (x) [2 points)
Let X be a discrete random variable with values in N = {1, 2,...}. Prove that X is geometric with parameter p = P(X = 1) if and only if the memoryless property P(X = n + m | X > n) = P(X = m) holds. To show that the memoryless property implies that X is geometric, you need to prove that the p.m.f. of X has to be P(X = k) = p(1 - p)^(k-1). For this, use...
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.
Let X ...., X., be a random sample from a geometric distribution with parameter , and let the prior distribution of be uniform on the interval from zero to one. 1. (70 Points) Find the density of the posterior distribution, including the constant that makes it integrate to one. Name the distribution.