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).
Find moment generating function of geometric distribution f(x)=p*q^(x-1), where x=1, 2, ... and use it to...
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.
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
MoM stands for Method of Moments. 4. (a) If X Geometric(p), prove that the moment-generating function for X is Mx(t) pe 1-(1-p)e' (b) Use your result of part (a) to show that E(X) = p and V(X) = Now, we have X1, X2,... X, d Geometricíp). (c) Find a MoM estimator for p based on the first moment. (d) Explain why your estimator makes sense intuitively. (e) Use the following data to give a point estimate of p: XnGeometric(p 3,...
Problems binomial random variable has the moment generating function ψ(t)-E( ur,+1-P)". Show, that EIX) np and Var(X)-np(1-P) using that EXI-v(0) and Elr_ 2. Lex X be uniformly distributed over (a b). Show that EX]- and Varm-ftT using the first and second moments of this random variable where the pdf of X is () Note that the nth i of a continuous random variable is defined as E (X%二z"f(z)dz. (z-p?expl- ]dr. ơ, Hint./ udv-w-frdu and r.e-//agu-VE. 3. Show that 4 The...
The moment generating function ф(t) of random variable X is defined for all values of t by et*p(x), if X is discrete e f (x)dx, if X is continus (a) Find the moment generating function of a Binomial random variable X with parameters n (the total number of trials) and p (the probability of success). (b) If X and Y are independent Binomial random variables with parameters (n1 p) and (n2, p), respectively, then what is the distribution of X...
2. Consider the Poisson distribution, which has a pdf defined as: a) Derive the moment generating function. b) Use the moment generating function and the method of moments to find the mean and the variance. c) If X follows the Poisson distribution with Xx - 2.3, and Y follows a Poisson distribution with XY-54, what is the distribution of the sum X + Y, assuming that X and Y are independent?
Derive the moment generating function of the binomial distribution and calculate the mean and variance. p(x)=(*)*(1+p)** x = 0,1,2,...,
problems binomial random, veriable has the moment generating function, y(t)=E eux 1. A nd+ 1-p)n. Show that EIX|-np and Var(X) np(1-p) using that EIX)-v(0) nd E.X2 =ψ (0). 2. Lex X be uniformly distributed over (a b). Show that ElXI 쌓 and Var(X) = (b and second moments of this random variable where the pdf of X is (x)N of a continuous randonn variable is defined as E[X"-广.nf(z)dz. )a using the first Note that the nth moment 3. Show that...
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...
Derive the moment generating function of the binomial distribution and calculate the mean and variance. P(x) = x = 0,1,2,...,