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 P(X = k) = P(X = k + 1|X > 1) repeatedly.
Let X be a discrete random variable with values in N = {1, 2,...}. Prove that...
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.
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.)
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...
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
Problem 3. Let X be a discrete random variable that takes values in N. Show that if X is memory-free then it must be the case that X Geo(p) for some p. (Hint a useful first step might be to show that P(X > t)= P(X > 1)' for all t E N.) Problem 3. Let X be a discrete random variable that takes values in N. Show that if X is memory-free then it must be the case that...
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)=...
Exercise 2.23 If X is a discrete random variable having the Poisson distribution with parameter that the probability that X is even is e cosh A. Exercise 2.24 If X is a discrete random variable having the geometric distribution with parameter p. show that the probability that X is greater than k is (1 -p)k à, show
2. Let X be an exponential random variable with rate A > 0. In this problem you will show that X satisfies the memoryless property. Let s 2 0 and t > 0. Show that P(X > t + s| X > s) = e-M
Let X be a discrete random variable taking integer values 1, 2, ..., 10. It is also known that: P(X < 4) = 0.57, PCX 2 4) = 0.71. Then P(X = 4) = A: 0.14|B: 0.28 |C: 0.45 OD: 0.64|E: 0.73 OF: 0.95 Submit Answer Tries 0/5