Let Xi’s are i.i.d Poisson(λ) random variables for 1 ≤ i ≤ n. Derive the distribution...
Let Xi and X2 be independent Poisson random variables with means λ! and λ2. (a) Find the distribution of x, + x, (b) Compute the conditional distribution of Xi given that X + X.-n
Problem 3: Suppose X1, X2, is a sequence of i.i.d. random variables having the Poisson distribution with mean λ. Let A,-X, (a) Is λη an unbiased estimator of λ? Explain your answer. (b) Is in a consistent estimator of A? Explain your answer 72
Let N be a Poisson random variable with parameter λ. Suppose ξ1,ξ2,... is a sequence of i.i.d. random variables with mean µ and variance σ2, independent of N. Let SN = ξ1 + ...ξN. Determine the mean and variance of SN
Let λ >0 and suppose that X1,X2,...,Xn be i.i.d. random variables with Xi∼Exp(λ). Find the PDF of X1+···+Xn. Use convolution formula and prove by induction
8. Let the random variables X be the sum of independent Poisson distributed random variables, i.e., X = -1 Xi, where Xi is Poisson distributed with mean 1. (a) Find the moment generating function of Xi. (b) Derive the moment generating function of X. (d) Hence, find the probability mass function of X.
5. Let X1, X2,... , X100 be i.i.d. random variables, following the normal distribution N(0, 102). Let α denote the probability that there are at least 3 variables among them whose absolute value is larger than 19.6. Compute α, and give an approxi- mate value of α with an error less than 0.01 according to the Poisson distribution. 15pts] 5. Let X1, X2,... , X100 be i.i.d. random variables, following the normal distribution N(0, 102). Let α denote the probability...
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3). 3. (25 pts.) Let X1,...
8. Let the random variables X be the sum of independent Poisson distributed random variables, i.e., X = 11-1Xị, where Xi is Poisson distributed with mean li. (a) Find the moment generating function of Xį. (b) Derive the moment generating function of X. (d) Hence, find the probability mass function of X.
Central Limit Theorem: let x1,x2,...,xn be I.I.D. random variables with E(xi)= U Var(xi)= (sigma)^2 defind Z= x1+x2+...+xn the distribution of Z converges to a gaussian distribution P(Z<=z)=1-Q((z-Uz)/(sigma)^2) Use MATLAB to prove the central limit theorem. To achieve this, you will need to generate N random variables (I.I.D. with the distribution of your choice) and show that the distribution of the sum approaches a Guassian distribution. Plot the distribution and matlab code. Hint: you may find the hist() function helpful
Q3 Suppose X1, X2, ..., Xn are i.i.d. Poisson random variables with expected value ). It is well-known that X is an unbiased estimator for l because I = E(X). 1. Show that X1+Xn is also an unbiased estimator for \. 2 2. Show that S2 (Xi-X) = is also an unbaised esimator for \. n-1 3. Find MSE(S2). (We will need two facts) E com/questions/2476527/variance-of-sample-variance) 2. Fact 2: For Poisson distribution, E[(X – u)4] 312 + 1. (See for...