Suppose two random variables Y1 and Y2 have the following quantities: E(Y) = 3, E(Y/2) =...
Consider two random variables with joint density fY1,Y2(y1,y2) =(2(1−y2) 0 ≤ y1 ≤ c,0 ≤ y2 ≤ c 0 otherwise (a) Find a value for c. (4 marks) (b) Derive the density function of Z = Y1Y2. (10 marks) . Consider two random variables with joint density fyiy(91, y2) = 2(1 - y2) 0<n<C,0<42 <c o otherwise (a) Find a value for c. (4 marks) (b) Derive the density function of Z=Y Y. (10 marks)
2. Suppose the variables Y1 and Y2 have the following properties: 2. Suppose the variables Yi and Y2 have the following properties: E(%) = 4, Var(%) = 19,E(%) = 6.5, Var(%) = 5.25,E(,%) = 30 Calculate the following; please show the underlying work: a) (3 pts) Cov(Y,Y2) b) (3 pts) Cov(4Y1,3Y2) c) (3 pts) Cov(4h, 5-½) d) (6 pts) Find the correlation coefficient between 1 + 3, and 3-2%
1. Suppose we have three random variables Y1 , Y2 , and Y3 . Suppose we have three random variables Y, Y,, and Y,. The standard deviations of Y and Y, are both 3 and the standard deviation of Y is 2. The correlation coefficient between Y and Y, is-0.6. The covariance between Y and Y, is 0.5. Y is independent of Y 1. 1 2 a) (3 pts) Find Var(h + 3%) b) (3 pts) Find Cov(3h + 2⅓'5½-%)
2. Let the random variables Y1 and Y, have joint density Ayſy22 - y2) 0<yi <1, 0 < y2 < 2 f(y1, y2) = { otherwise Stom.vn) = { isiml2 –») 05451,05 ms one a independent, amits your respon a) Are Y1 and Y2 independent? Justify your response. b) Find P(Y1Y2 < 0.5). on the
Let Y1 and Y2 be two independent discrete random variables such that: p1(y1) = 1/3; y1 = -2 ,- 1, 0 p2(y2) = 1/2; y2 = 1, 6 Let K = Y1 + Y2 a) FInd the moment Generating function of Y1, Y2, and K b) find the probability mass function of K
4. Suppose that X and Y are random variables with E(X) = 2, E(Y) = 1. E(X*) = 5, E(Y2-10, and E(XY) = 1 (a) Compute Corr(X,Y) (b) Choose a number c so that X and X +cY are uncorrelated
Suppose Y1, Y2, ... Yn are mutually independent random variables with Y1 ~ N(μ1, (σ1)^2) Y2 ~ N(μ2, (σ2)^2) ... Yn ~ N(μn, (σn)^2) Find the distribution of U=summation(from i=1 to n) ((Yi - μi)/σi)^2 I am not sure where should I start this question, could you please show me the detail that how you do these two parts? thanks :)
Suppose Y1, Y2, …, Yn are independent and identically distributed random variables from a uniform distribution on [0,k]. a. Determine the density of Y(n) = max(Y1, Y2, …, Yn). b. Compute the bias of the estimator k = Y(n) for estimating k.
2. Suppose that Y and Y2 are continuous random variables with the joint probability density function (joint pdf) a) Find k so that this is a proper joint pdf. b) Find the joint cumulative distribution function (joint cdf), FV1,y2)-POİ уг). Be y, sure it is completely specified! c) Find P(, 0.5% 0.25). d) Find P (n 292). e) Find EDY/ . f) Find the marginal distributions fiv,) and f2(/2). g) Find EM] and E[y]. h) Find the covariance between Y1...
3. (30pt) Suppose that E(Y) = 1, E(Y2) = 2, E(Y3) = 3, V(Y1) = 6, V(Y2) = 7,V (Y3) = 8, Cov(Yı, Y2) = 0, Cov(Yı, Y3) = -4 and 10 1 2 3 Cov(Y2, Y3) = 5. Also define a = 20 and A = 4 5 6 30/ ( 7 8 9 (a) (10pt) Find the expected value and variance covariance matrix of Y, where Y = Y2 (b) (10pt) Compute Eſa'Y) and E(AY). (c) (10pt) Compute...