MoM stands for Method of Moments.
MoM stands for Method of Moments. 4. (a) If X Geometric(p), prove that the moment-generating function...
LetX1,...,Xn be a random sample of size n from the geometric distribution for which p is the probability of success. (a) Use the method of moments to find a point estimator for p. (b) Use the following data (simulated from geometric distribution) to find the moment estimator for p: 2 5 7 43 18 19 16 11 22 4 34 19 21 23 6 21 7 12 The pdf of a geometric distribution is f(x)= p(1-p)^x-1, for x,.... Also population...
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).
The geometric random variable X has moment generating function given by EetX) = p(1 – qe*)-7, where q = 1- p and 0 < p < 1. Use this to derive the mean and variance of X.
5.2.1. Let XX be a random sample of size n from the geometric distribution for which p is the probability of success. (a) Use the method of moments to find a point estimator for p. (b) Use the following data (simulated from geometric distribution) to find the moment estimator for p: 2 5743 18 19 16 11 22 4 34 19 21 23 6 21 7 12
Problem 4 (10 Points) Derive the moment generating function and the characteristic function for a geometric random variable X with parameter p. Compute E(X) for p = 4/5. Prohlom 510 Pointe Tot Y Y Y be indarrandort randomiblog ebore V Tot
12. let Mx(1) be the moment generating function of X. Show that (a) Mex+o(t) = eMx(at). (b) TX - Normal(), o?) and moment generating function of X is Mx (0) - to'p. Show that the random variable 2 - Normal(0,1) 13. IX. X X . are mutually independent normal random variables with means t o ... and variances o, o,...,0, then prove that X NOEL ?). 14. If Mx(1) be the moment generating function of X. Show that (a) log(Mx...
The moment generating function (MGF) for a random variable X is: Mx (t) = E[e'X]. Onc useful property of moment generating functions is that they make it relatively casy to compute weighted sums of independent random variables: Z=aX+BY M26) - Mx(at)My (Bt). (A) Derive the MGF for a Poisson random variable X with parameter 1. (B) Let X be a Poisson random variable with parameter 1, as above, and let y be a Poisson random variable with parameter y. X...
9. (9 pts) The random variable r-Gamma(x-2, β-4). functions to prove that the moment generating function for the random variable W mw(t) (1-12t)2. Use the method of moment-generating 3Y +5is eSt 10, (9 pts) Suppose that Y has a gamma distribution with α-n/2 for some positive integer n and β equal to some specified value. Use the method of moment-generating functions to prove that W- 2Y /g has a Chi-squared distribution with n degrees of freedom. Make sure you show...
4. The moment generating function of the normal distribution with parameters μ and σ2 is (t) exp ( μ1+ σ2t2 ) for -oo < t oo. Show that E X)-ψ(0)-μ and Var(X)-ψ"(0)-[ty(0)12-σ2. 5. Suppose that X1, X2, and X3 are independent random variables such that E[X]0 and ElX 1 for i-12,3. Find the value of E[LX? (2X1 X3)2] 6. Suppose that X and Y are random variables such that Var(X)-Var(Y)-2 and Cov(X, Y)- 1. Find the value of Var(3X -...
1. A binomial random variable has the moment generating function, (t) E(etx)II1 E(etX) (pet+1-p)". Show that EX] = np and Var(X) = np(1-p) using that EX] = ψ(0) and E(X2] = ψ"(0). 2. Lex X be uniformly distributed over (a,b). Show that E[xt and Var(X) using the first and second moments of this random variable where the pdf of X is f(x). Note that the nth moment of a continuous random variable is defined as EXj-Γοχ"f(x)dx (b-a)2 exp 2