ility mass function of X and Y, p(x, y) is g p(2,1)= p(3,1)= } 9 p(2,2)=0 p(3,2)=+ p(1,1)= p(1,2)=
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- b) Let Y1, Y2, ..., Yn be a random sample from a population with probability density function in part a). Show that the best test for the hypothesis in part a) rejects Ho if |yi <c i=1 where c solves the probability equation a = P(II1Yis c[0 = 2). c) Let X1,X2, ..., Xn be a random sample from GAMMA(2,ß) distribution, and consider Y = E-,Xi- Show whether or not Y is a pivotal quantity and give its distribution.2 cards are chosen at random from a deck of 52 cards. Suppose we win $2 for each spade and lose $1 for each heart or diamond selected, we neither win nor lose if we pick a club. Let X denote our winnings and px be the probability mass function of . What is px(2)?1)A random vector z = (x y) T has joint probability density given by:a)Sketch the graph of the probability density function pxy(X, Y ).b)Determine the value of the constant C.c)Determine the mean mz vector.
- 1) Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the estimator of moments for the parameter θ.The probability mass function for a discrete random variable X is defined as ((1+0)-(x) 0; x = 0,1,2,3,..., η (x) = {(1+0) (2) *; fx(x) 0; e.w. where 0 > 0. Show that it is probability mass function. Find its mean and variance.A penny and a nickel are tossed. The penny has probability 0.4 of coming up heads, and the nickel has probability 0.6 of coming up heads. Let X = 1 if the penny comes up heads, and let X = 0 if the penny comes up tails. Let Y = 1 if the nickel comes up heads, and let Y = 0 if the nickel comes up tails. a) Find the probability mass function of X. b) Find the probability mass function of Y. c) Is it reasonable to assume that X and Y are independent? Why? d) Find the joint probability mass function of X and Y.
- Two sharpshooters will simultaneously shoot once at a target. Let Xi , i = 1, 2, be marksman i's horizontal error, measured in meters. If the target's position is (a, b), and marksman i's shot ends up at (x, y), then Xi = (x - a). These errors are modeled by X1 ~ N(0, 0.5) and X2 ~ N(0, 1). What is the probability that both shots will be within 1 meter of the target, in the horizontal direction?5)Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find a sufficient statistics for θ. Check if the statistics you have found are minimally sufficient.The joint probability mass function of the two random variables (X, Y) is given by f(x,y) = { 1/5(3x-y), 1<=x<=2 , 1<=y<=3 0 , otherwise• Find the P( X ≤ Y)• Find the marginal density functions of X and Y• Are X and Y independent?• Find E(XY)
- Let X and Y be independent Gaussian random variables, each distributed according to (0, σ2). 1. Find the joint density function of the random variables Z = X + Y and W = 2X −Y . What is the correlation coefficient between these two random variablesA random variable x has a probability mass function defined by P(X=x) = r(x-1). if x= 2,3 r(8-x). if x=4,5,6 0. Otherwise Write out the probability mass function. What will be the expectation and variance of x. Find P (x greater or equal to 2 and less or equal to 5).Two archers will simultaneously shoot once at a target. Let Xi , i = 1, 2, be marksman i's horizontal error, measured in meters. If the target's position is (a, b), and marksman i's shot ends up at (x, y), then Xi = (x - a). These errors are modeled by X1 ~ N(0, 0.5) and X2 ~ N(0, 1). What is the probability that both shots will be within 1 meter of the target, in the horizontal direction?
