Let ?(?, ?) = 3 2 ? 2?, 0 ≤ ? ≤ 1, 0 ≤ ? ≤ 2
a. Compute the marginal probability mass functions for ? and ?
b. Are ? and ? independent? Why?
c. Compute ??, ?? 2 , ??, and ?? 2 .
d. Compute ???(?, ?) and the correlation coefficient
e. Find ?(? > ?)
(1 point) 3. Let X and Y be random variables with a joint probability density function f(z, y)e (a)Find the marginal distribution functions of X and Y, respectively. i.e. Find f(z) and f(y) f(x)- elsewhere (b) Identify the distribution of Y. What is the E(Y) and SD(Y) E(Y)- (c) Are X and Y independent random variables? Show why, or why not (d) Find P(1 X 2|Y 1) E SD(Y)-
1 * Consider the following joint distribution for the weather in two consecutive days. Let X and Y be the random variables for the weather in the first and the second days, whereas the weather is coded as 0 for sunny, 1 for cloudy, and 2 for rainy. 0 0.2 0.2 0.2 10.1 0.1 0.1 2 0 0.1 0 (a) Find the marginal probability mass functions for X and Y (b) Are the weather in two consecutive days independent? (c)...
5. (40 points) Let f(x,y) = (x + y),0 < 2,2 <y < 1 be the joint pdf of X and Y. (1) Find the marginal probability density functions fx(x) and fy(y). (2) Find the means hx and my. (3) Find P(X>01Y > 0.5). (4) Find the correlation coefficient p.
Let X1 and X2 be two discrete random variables, where X1 can attain values 1, 2, and 3, and X2 can attain values 2, 3 and 4. The joint probability mass function of these two random variables are given in the table below: X2 X1 2 3 4 1 0.05 0.04 0.06 2 0.1 0.15 0.2 3 0.2 0.1 0.1 a. Find the marginal probability mass functions fX1 (s) and fX2 (t). b. What is the expected values of X1...
Let X and Y be two independent Bernoulli (05) randon variables and define U = X + Y and (a) Find the joint and marginal probability mass functions for U and V. [It is sufficient to con struct a table to describe these mass functions.] (b) Are U and V independent? Why or why not? (c) Find the conditional probability mass functions pUv (u) and pv u(v). [Again, you can construct a table to describe these mass functions.]
4. (30 pts) Let (X,Y) have joint pdf given by < , | e-9, 0 < x < f(x,y) = 3 | 0, 0.w., (a) Find the correlation coefficient px,y: (20 pts) (b) Are X and Y independent? Explain why. (10 pts)
2 1 -2 3 0 1 4 2 1. Let B -3 0 3 ( 1) 2 2 -1 0 (a) Find det(B).(Show all work.) -3 -R2- .A 4 O0-2/2 1-3 0 3 入ス-1 0 I-2 3 det ao -1 O 3 1-3 RyR-( 2 2-10 420 4 (b) Find det(BT). (c) Find det(B-1). (d) Find det(-B) . (e) Is 0 an eigenvalue of B? (f) Are thè columns of B linearly independent?
Let X and Y have the joint pdf f(x,y) = e-x-y I(x > 0,y > 0). a. What are the marginal pdfs of X and Y ? Are X and Y independent? Why? b. Please calculate the cumulative distribution functions for X and Y, that is, find F(x) and F(y). c. Let Z = max(X,Y), please compute P(Z ≤ a) = P(max(X,Y) ≤ a) for a > 0. Then compute the pdf of Z.
Problem 2: (8 points) Let X be the number of hoses being used on the self-service and Y that being used on the full-service on an Island. The joint p.m.f. of X and Y is given by 1 y 0 2 0 0.1 0.04 0.02 0.08 0.2 0.06 2 0.06 0.14 0.3 1 (a) Show that this is a valid joint p.m.f. [1] (b) Fill out the table with the marginal distributions of X and Y. [2] (c) Calculate E(X),...
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...