Suppose \( X \) is a random variable taking possible values in \(\{1, 2, 3, \ldots\}\) and \(0 < P(X = 1) < 1\), and that \( X \) satisfies the memoryless property. Prove that \( X \) must be a geometrically distributed random variable for some parameter value \( p \).

Let X be the random variable having possible values {1,2,3...} and 0<p<1X satisfies the…

Answered: Suppose X is a random variable taking…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

See similar textbooks

Similar questions

Let X, Y, and Z be independent uniform(0, 1) random variables. (a) Find P(X/Y ≤t) and P(XY ≤t). (b) Find P(XY/Z ≤ t).
= 3. Propose an algorithm for simulating the occurrence of two dependent events A and B that uses only one uniform random variable U~Unif(0, 1). Assume that the probabilities P(A) ΡΑ, P(B) = PB, and P(AUB) = q are given, where PA, PB, q = (0, 1) and max(PA, PB) < q < PA+PB· Formulate your answer in the form of the following statement: "if U E ..., then event A occurs; if U € .……., then event B occurs."
Suppose X is a discrete random variable. Let the pmf of X be equal to 5 - x f(x) = x = 1,2,3,4 10 Find the cdf of X, that is F(X).
2. Let X₁, X2, X3,..., be independent identically distributed Unif(0, 1) random variables. Let Sn be the largest gap between any two of the points {X1, X2,..., Xn} (e.g. if n = 3 and (X₁, X2, X3) = (0.1, 0.5, 0.7) then S3 = 0.4). Show that for any x > 0 lim P(nSn > x) = 1. n→∞ Hint: to lower bound nSn > x, if suffices to demonstrate that there is an empty interval of some length x/n which contains none of the points. Make a list of some candidate intervals and then use Cheby- shev's inequality to show one of them is empty with probability going to 1.
Let X1 and X2 be discrete random variables with joint pdf given by the following table: X2 1 2 3 1 1 1 12 2 1 X1 3 1 2 18 4 15 a) Find the marginal pdf's of X, and X2 b) Are X, and X2 independent? Why or why not? c) Find P(X1 < X2)
2. Suppose one has n non-degenerate random variables X1, X2,, X, so that X1+ X2 + · · · + Xn = L for some constant L. (Recall that a random variable is non-degenerate if it is not a constant in disguise.) Show that there must be at least one pair of indices i j so that p(X;, X;) < 0.

Recommended textbooks for you

A First Course in Probability (10th Edition)

Probability

ISBN:

9780134753119

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability

Probability

ISBN:

9780321794772

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability (10th Edition)

Probability

ISBN:

9780134753119

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability

Probability

ISBN:

9780321794772

Author:

Sheldon Ross

Publisher:

PEARSON

SEE MORE TEXTBOOKS

Suppose X is a random variable taking possible values in {1, 2, 3,... } and 0 < P(X = 1) < 1, and that X satisfies the memoryless property. Prove that X must be a geometrically distributed random variable for some parameter value p.