here as we know that mean E(X)=xP(x)=1*(1/k)+2*(1/k)+3*(1/k)+4*(1/k)+...k*(1/k)
E(X)=(1/k)*(1+2+3+4+...k)=(1/k)*(k(k+1))/2
E(X)=(k+1)/2
E(X2)=x2P(x)=(1/k)*(12+22+32+..+k2)=(1/k)*(k(k+1)(2k+1))/6
=(k+1)*(2k+1)/6
therefore Variance =E(X2)-(E(X))2 =(k+1)*(2k+1)/6-(k+1)2/4
3.18. Find the mean and variance of the given PMF pr)-1/k, where - 1,2,3,, k.
4. The radius of a sphere is a discrete random variable with pmf given by: PR()1,2,3 0 otherwise (a) Find k (b) Find the mean of the VOLUME of the sphere (c) Find the variance of the VOLUME of the sphere. (d) Find the mean and the variance of the sample mean of an iid sample contag six spheres.
3. Let (X, Y) be a bivariate random variable with joint pmf given by x= 1,2,3, y = 0,1,2,3, ... ,00 f(x, y) 12 0 e.w. (a) Show that f(x, y) is a valid joint pmf. (b) Find fa(x) (i.e. the marginal pmf of X). (c) Find fy(y) (i.e. the marginal pmf of Y). (d) Find P [Y X]
3. (6 pts) Let Z be standard normal,(mean-0, variance-1) (a) Find Pr(Z1.13)
2. Let Px(x) = 1, X = 1,2,3, 4, 5, zero elsewhere, be the pmf of X. Find P(X = 1 or 2), P(3 < X < ), and P(1 < X < 2).
If E(Xr) = 6, r = 1,2,3, , find the moment generating function M(t) of X ạnd the pmf, the mean, and the variance of X ( M(t)-Σ000 M !(0) origin) rk, and note that Mrk, (0) = ElXkl is the kth moment of X about the
Let the joint pmf of X and Y be p(x, у) схуг, x-1,2,3, y-12. a) Find constant c that makes p(x, y) a valid joint pmf. c) Are X and Y independent? Justify d) Find P(X+Y> 3) and PCIX-YI # 1)
2. 6. (20points). Let X and Y have the joint pmf ? 1,2,3 , zero elsewhere. (a) Find the mgf M(ti, 2) of this joint distribution. (b) Compute the means and the variances, and the correlation coefficient of X and Y (c) Determine the conditional mean E(Xly)-(extra point)
3. (a) Let (X,Y) have the joint pmf (2 + y + k – 1)! P(X = 1, Y = y) => pip (1- P1 - p2), r!y!(k − 1)! where r, y=0,1,2, ..., k> 1 is an integer, 0 <P1 <1,0 <p2 <1, and p1 + P2 <1, find the marginal pmfs of X and Y and the conditional pmf of Y given X = r.
(10 points) One observation is taken on a discrete random variable X with pmf: f(;), where 1. 0 E 1,2,3. Find the MLE of 0 0 0 2 0
2. Let X ~ Bin(4, ), i.e., the PMF of X is given by Px(1) = (1) (1) *,* = 0,1,2,3,4. Find Px B(k) where B = {X #0}.