Let M be a Poisson (λ) random variable having M equal m. If we flip a p-biased coin m times and let X be the number of heads, show that X is a Poisson (pλ) random variable. Use the identity for k= 0 to infinity Σy^k/k! =e^y
Let M be a Poisson (λ) random variable having M equal m. If we flip a...
Show all details: Exercise 10.4. Let X be a Poisson random variable with parameter λ. That is, P(X = k) e-λλk/kl, k 0.1 Compute the characteristic function of (X-λ)/VA and find its limit as Exercise 10.4. Let X be a Poisson random variable with parameter λ. That is, P(X = k) e-λλk/kl, k 0.1 Compute the characteristic function of (X-λ)/VA and find its limit as
Suppose you flip a coin 15 times and let x be the discrete random variable of the number of heads obtained. Use the binomial distribution table to find each of the following probabilities. (A) p(exactly 8 heads)= (b) p(at least one head)= (c) P(at most 3 heads)=
Let X be a Poisson random variable with parameter λ = 6, and let Y = min(X, 12). (a) What is the p.m.f. of X? (b) What is the mean of X? (c) What is the variance of X? (d) What is the p.m.f. of Y? (e) Compute E[Y ].
probability: please solve it step by step. thanks An unfair coin has probability of heads equal to p. An experiment consists of flipping this unfair coin n times and then counting the number of heads. a. Let Y; be a random variable which is 1 if the ith flip is heads and 0 if the ith flip is tails, where 1 sisn. Show that E (Y) = p and V(Y) = p-p2. b. Derive the moment-generating function of Y. c....
Problem 4 (Poisson Distributed RV) In a composite experiment, Z is a Poisson-distributed RV with parameter A, i.e P(Z k, and a coin with probability p of showing heads is tossed Z times (after observing Z). Let X denote the number of heads and Y denote the number of tails after Z tosses (a) Clearly specify the sample space 2xy, the set of values taken jointly by X and Y. (b) Determine the joint pmf of X and Y (in...
5. Let X be a Poisson random variable with parameter λ = 6, and let Y = min(X, 12). (a) What is the p.m.f. of X? (b) What is the mean of X? (c) What is the variance of X? (d) What is the p.m.f. of Y? (e) Compute EY
Coin with random bias. Let P be a random variable distributed uniformly over [0, 1]. A coin with (random) bias P (i.e., Pr[H] = P) is flipped three times. Assume that the value of P does not change during the sequence of tosses. a. What is the probability that all three flips are heads? b. Find the probability that the second flip is heads given that the first flip is heads. c. Is the second flip independent of the first...
Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X is r E 0,1,2,...) This distribution is often used to model the number of events which will occur in a given time span, given that λ such events occur on average a) Prove by direct computation that the mean of...
You flip a coin 100 times. Let X= the number of heads in 100 flips. Assume we don’t know the probability, p, the coin lands on heads (we don’t know its a fair coin). So, let Y be distributed uniformly on the interval [0,1]. Assume the value of Y = the probability that the coin lands on heads. So, we are given Y is uniformly distributed on [0,1] and X given Y=p is binomially distributed on (100,p). Find E(X) and...
c) d) 120 200 10) We flip a fair coin 4 times. Define a random variable X = number of heads we obtain. Thus X=0,1,2,3, or 4 If p(x) denotes the probability function for X, find p(3). a) 1/16 b) 2/16 c) 3/16 d) 4/16 5/16