1. Find p such that P( X > 2 ) = 1/2 where X is Geometric( p )

Find moment generating function of geometric distribution f(x)=p*q^(x-1), where x=1, 2, ... and use it to find EX and DX (i.e. find the first and the second moments).
1) let X follows a geometric distribution, Geo(p). Find P(X=an even number). 2) let X follows a geometric distribution. For positive integers, n, m, show that a). P(X>n) = (1-p)^n b). P(X>n+m|X>n) = (1-p)^m = P(X>m). hint: this property is called the memory-less property of the geometric distribution.
K = 4
Let X → Geometric (p) where p such that P (X > n) =t. and k is number of letters in your j
Let X → Geometric (p) where p such that P (X > n) =t. and k is number of letters in your j
Let X, Y Geometric(p) be independent, and let Z a. Find the range of 2. b. Find the PMF of Z c. Find EZ.
Let X, Y Geometric(p) be independent, and let Z a. Find the range of 2. b. Find the PMF of Z c. Find EZ.
Let X ? Geometric(p) with probability mass function P(X =x)=p(1?p)x?1, x?N. (a) Verify that FX (x) = 1 ? (1 ? p)x, x ? {1, 2, 3, . . .}. (b) Graph FX(x) for x ? [?1,4] for p = 1/4. (c) Let X ?Geometric(1/4). Find P(X ? (3, 5]) and P(X is even).
8. (8 points) Let X1, X2, . . . , X, bea random sample from the geometric distribution with pmf f(aip) (1-P)-p,z1,2,3,..., where 0 <ps 1. Find the maximam likel ihood estimator of p and show that the maximum likelthood estimator is unblased.
8. (8 points) Let X1, X2, . . . , X, bea random sample from the geometric distribution with pmf f(aip) (1-P)-p,z1,2,3,..., where 0
Recall the Geometric(p) distribution where X = number of flips of a coin until you get a head (H) with Pr(H) = p. The distribution is Pr(X = x) = (1 − p) (x−1) p for x = 1, 2, . . . , with mean E(X) = ∑ x=1∞ (x(1 − p) (x−1) p) = 1/p, which can be obtained by brute force. An easier way to find the mean is to condition on the first toss, say Y...
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).
Problem 0.2 Recall the Geometric(p) distribution where X-number of flips of a coin until you get a head (H) with Pr(H) -p. The distribution is Pr(X- (1-p)1p for 1,2,. , with mean E(X)x(1 - p)*-p- 1/p, which can be obtained by brute force. An easier way to find the mean is to condition on the first toss, say Y- 0or 1 if the first toss is T or H. Show the mean is 1/p using E(X) EE(X Y)
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