Please Answer both part a and b clearly.
Please Answer both part a and b clearly. U(0, 1). Find the pdf of Y =...
Let X and Y ~U(0, 1]. X and Y are independent a) Find the PDF of X+Y b) Suppose now X~(0, a] Y~(0,b] and . Find the PDF of X+Y Ο <α<b
6. Suppose X and Y have the joint pdf fr,y) = 2 exp(-:- 0 ) 0< <y otherwise o a. Find Px.x, the correlation coefficient between X and Y. b. Let U = 2X-1 and V=Y +2. What is pu.v, the correlation coefficient between U and V? c. Repeat (b) if U = -TX and V = Y + In 2. d. Let W = Y - X. Compute Var (W). e. Refer to (d). Find an interval that will...
(15 points) Consider two independent, exponential random variables X,Y ~ exp(1). Let U = X + Y and V = X/(X+Y). (a) (5 points) Calculate the joint pdf of U and V. (b) (5 points) Identify the distribution of U. If it has a "named” distribution, you must state it. Otherwise support and pdf is enough. (c) (5 points) Identify the distribution of V.If it has a "named” distribution, you must state it. Otherwise support and pdf is enough.
Consider the joint pdf of the random variables X and Y : 1/8, if 0 ≤ y ≤ 4, y ≤ x ≤ y + 2 f (x, y) = 0, otherwise (i) Draw the region where f (x, y) ̸= 0. Shade its area. (ii) Compute the probability P (X + Y ≤ 2). (iii) Compute the marginal pdf f1(x) of X. Specify clearly its support, i.e., the subset of the real line such that f1(x) ̸= 0. (iv)...
exp(1) 7. (15 points) Consider two independent, exponential random variables X,Y Let U = X + Y and V = X/(X+Y). (a) (5 points) Calculate the joint pdf of U and V. (b) (5 points) Identify the distribution of U. If it has a "named" distribution, you must state it. Otherwise support and pdf is enough. (c) (5 points) Identify the distribution of V.If it has a “named” distribution, you must state it. Otherwise support and pdf is enough.
5. Random variables X U[0, 1 and Y ~Exp(1) are independent (a) Compute P(X Y > z) for the case 0 S1 and the case z >1. b) Compute and plot the pdf of XY. (c) Give the MGF of X Y.
5. Random variables X U[0, 1 and Y ~Exp(1) are independent (a) Compute P(X Y > z) for the case 0 S1 and the case z >1. b) Compute and plot the pdf of XY. (c) Give the...
The joint probability density function (pdf) of (X,Y ) is given by f(X,Y )(x,y) = 12/ 7 x(x + y), for 0 ≤ y ≤ 1, 0 ≤ x ≤ 1, 0, elsewhere. (a) Find the cumulative distribution function of (X,Y ). Make sure you derive expressions for the cdf in the regions • x < 0 or y < 0; • 0 ≤ x ≤ 1, 0 ≤ y ≤ 1; • x > 1, 0 ≤ y ≤...
Let the joint pdf of X and Y be , zero elsewhere. Let U = min(X, Y ) and V = max(X, Y ). Find the joint pdf of U and V . 12 (x+y), 0< <1,0 y<1 f (x, y) 12 (x+y), 0
I don't understand a iii and b ii, What's the procedure of
deriving the limit distribution? Thanks.
6. Extreme values are of central importance in risk management and the following two questions provide the fundamental tool used in the extreme value theory. (a) Let Xi,... , Xn be independent identically distributed (i. i. d.) exp (1) random variables and define max(Xi,..., Xn) (i) Find the cumulative distribution of Zn (ii) Calculate the cumulative distribution of Vn -Zn - Inn (iii)...
That is, the distribution of X has pdf given by θ-11(1 < x 0) and a point mass on {x-1). (b) Let X1, X2,..., Xn be a random sample from the distribution in part (a). Show that the pdf of the maximum order statistic X(n) is given by JX(n) (c) Show that X(n) is a sufficient statistic for θ. Is X(n) complete?