(a)
At time t = m, total number of trials is t/ = m / = m
By CDF of geometric distribution, probability of no success till m trials is,
(b)
Let T be the time of first success. Then the probability that no success has been observed by time t is,
As, and
{}
{t = m}
{ }
(c)
From part (b),
where F(t) is the CDF of T.
PDF of t is,
which is the PDF of exponential distribution with the parameter .
Thus, the time of first success T is approximately an exponential distribution variable with parameter .
Problem 2. This problem investigates the similarity between the geometric and ex- ponential random variables observed...
We have seen that the geometric distribution Geo(p) is used to model a random variable, X that records the trial number at which the first success isachieved after consecutive failures in each of the preceding trials ("success" and failure being used in a very loose sense here). Here, p is the success probability in each trial. We described the geometric distribution using the probability mass function: f(X)(1- p)*-1p, which computes the probability of achieving success in the xth trial after...
Prove that Box-Muller method described in class generates independent standard normal random variables. 4 a) Prove that the Box-Muller method described in class generates independent standard ll generate n to write a function which wi σ-) random variables b) Suppose that X is an exponential random variable with rate parameter λ and that Y is the integer part of X. Show that Y has a geometric distribution and use this result to give an algorithm to generate a random sample...
5. Develop an acceptance-rejection technique for generating a geometric random variable, X, with parameter p on the range {0,1,2, ...} . (Hint: X can be thought of as the number of trials before the first success occurs in a sequence of independent Bernoulli trials.)
Suppose T is a continuous random variable whose probability is determined by the ex- ponential distribution, f(t), with mean μ. a. Compute the probability that T is less than p b. The median of a continuous random variable T is defined to be the number, m, such that P(T which mIn other words, if f(t) is the PDF of T, it is the number m for P(T )f(t) dt Compute the median for the exponential random variable T above. Is...
Problem 5 (10 points). Suppose that the independent Bernoulli trials each with success probability p, are performed independently until the first success occurs, Let Y be the number of trials that are failure. (1) Find the possible values of Y and the probability mass function of Y. (2) Use the relationship between Y and the random variable with a geometric distribution with parameter p to find E(Y) and Var(Y).
4 a) Prove that the Box-Muller method described in class generates independent standard ll generate n to write a function which wi σ-) random variables b) Suppose that X is an exponential random variable with rate parameter λ and that Y is the integer part of X. Show that Y has a geometric distribution and use this result to give an algorithm to generate a random sample of size n from the geometric distribution with specified success probability p implementing...
Problem 2. Suppose a website sells X computers where X is modeled as a geometric random variable with parameter pi. Suppose that each computer is defective (i.e., needs to be returned for repair or replacement). independently with probability p2. Let Y be the mumber of computers sold which are defective. For this problem, recall that a geometric random variable X with parameter pi has pmf otherwise (a) Find ElY. (b) Find Var(Y). (c) Find P(Y 0).
Basic Probability Let us consider a sequence of Bernoulli trials with probability of success p. Such a sequence is observed until the first success occurs. We denote by X the random variable (r.v.), which gives the trial number on which the first success occurs. This way, the probability mass function (pmf) is given by Px(x) = (1 – p)?-?p which means that will be x 1 failures before the occurrence of the first success at the x-th trial. The r.v....
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
2. (Ross 3.2) Let Xi and X2 be independent geometric random variables having the same parameter p. (a) Compute the pmf for the random variable Y (b) Compute Pr(X,-iX, +X2=n) - Xi+ X2