2. Suppose that Z, and Z, have a joint probability density function given byG1 0s z, 51, 0sz s1- 2,2),f(Z,Z2) = 30,elsewherea) What is the marginal distribution of Z,.b) Find P(Z, s)

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Answered: 2. Suppose that Z, and Z, have a joint…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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i)only Let (X1,...,Xn), n ≥ 2, be a random sample from a distribution with discrete probability density fθ,j,whereθ ∈ (0,1), j = 1, 2, fθ,1 is the Poisson distribution with mean θ,andfθ,2 is the binomial distribution with size 1 and probability θ. (i) Show that T = ni=1Xi is not sufficient for (θ,j).
Suppose the random variables X and Y have joint probability density function f(x,y) given by: (image)Find: P(X < Y) = fX|Y=y (x)
Suppose that X is a continuous random variable with density function f(x). If f(x)=k for −5≤x≤3 and f(x)=0 otherwise, determine the value of k.
Let (X1,...,Xn), n ≥ 2, be a random sample from a distribution with discrete probability density fθ,j,whereθ ∈ (0,1), j = 1, 2, fθ,1 is the Poisson distribution with mean θ,andfθ,2 is the binomial distribution with size 1 and probability θ. (i) Show that T = ni=1Xi is not sufficient for (θ,j).
Suppose that the random variables X and Y have a joint density function given by: f(x,y) = {c(2x+y) for 2≤x≤6 and 0≤y≤5, 0 otherwise P(3 < X < 5, Y >1), P(X < 3), P(X +Y > 5), Find the joint distribution function (cdf),
Suppose that X, Y , and Z are random variables with a joint density f(x, y, z) = ( 6/((1+x+y+z)^4)) , when x, y, z > 0, and 0, otherwise. Determine the distribution of X + Y + Z.
A point D is chosen on the line AB, whose midpointis C and whose length is a. If X, the distance from D toA, is a random variable having the uniform density withα = 0 and β = a, what is the probability that AD, BD,and AC will form a triangle?
Suppose a continuous random variable X~Fx(x): f(x,y) = {1/4e^-1x/4, if x≥0 0, x<0} What is the cumulative density function of Y=min{2,X}?
Suppose that two-dimensional continuous random variable (X, Y) has joint probability density function given by f(x,y) = 24xy, x is less than equal to 1 and greater than equal to 0, y is less than equal to 1 and greater than equal to 0, x+y is less than equal to 1 and greater than equal to 0. Check that E(Y) = E[E(Y|X)] and V(Y) = E[V(Y|X)] + V[E(Y|X)].
X and Y are continuous random variables with pdf f(x,y) = 2x for0 ≤x ≤y ≤1, and f(x,y) = 0 otherwise. Find the conditional expectation ofY given X = x.
Suppose that X and Y are independent and uniformly distributed random variables. Range for X is (−1, 1) and for Y is (0, 1). Define a new random variable U = XY, then find the probability density function of this new random variable.
If X is a continuous random variable find the CDF and density of the function y = x/3

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2. Suppose that Z, and Z, have a joint probability density function given by f(Z,,2) =3(1- z,2), Osz s1, 0sz51 2,z2), 0, elsewhere a) What is the marginal distribution of Z,. b) Find P(Z, s)