pd.f. f given byf(x) = λe-d-a), X 1. Let Y be a rv. with M(t) for...
13. Let X,Y be rv's with pd.f. f given byf(x,y)-'e, x>0. y>o. Calculate the following quantities
Let X and Y be independent exponential random variables with pdfs f(x) = λe-λx (x > 0) and f(y) = µe-µy (y > 0) respectively. (i) Let Z = min(X, Y ). Find f(z), E(Z), and Var(Z). (ii) Let W = max(X, Y ). Find f(w) (it is not an exponential pdf). (iii) Find E(W) (there are two methods - one does not require further integration). (iv) Find Cov(Z,W). (v) Find Var(W).
5. Let F-(2x3y4 + 22, 2r4y3 + y). Given that f(x,y) rV + 2 2 + уг is a potential function of F. Find scVf dF, where C is a curve defined by() ,y int/2),0 st2.
5. Let F-(2x3y4 + 22, 2r4y3 + y). Given that f(x,y) rV + 2 2 + уг is a potential function of F. Find scVf dF, where C is a curve defined by() ,y int/2),0 st2.
Let the RV Y has the pdf f ( y ) = 6 y ( 1 − y ) , 0 ≤ y ≤ 1 , f ( y ) = 0 elsewhere . Find E[Y2]
4. Let X1, X, be two r.v.'s with m.g.f. given by t +t 9 12 ' Calculate E(K), σ(K) and C(X1 , X2), provided they are finite.
Suppose that a rv Y has mgf m(t)- (a) 1-bt) Differentiate this mgf twice and thereby obtain the mean and variance of Y. [5 marksj] (b) Suppose m(t) is the mgf of a rv W. Let r(t) be the natural logarithm of m(t), ie·r(t) = login(1). Find r'() and r"(t), and express r'(0) and r"(0) in terms of EW and VarW. [5 marks] Use the result in (b) to find the mean (d) Find the mean and variance of the...
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f) = J0 [f(t)]2dt. Prove that Ψ : X → R is differentiable. and find Ψ(f).
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X,...
For this question, let S be a sample space, and let RV be the set of {0, 1}-valued random variables. Let F : RV → (2^S) be given by F(X) := (X = 1). Let I : (2^S) → RV be the function that outputs the indicator variable for A on input A. Show that I and F are two-sided inverses. Note: 2^S denotes power set of S
The joint pdf for rv X, Y is given as follows:
if
1 ? x,y ? 2 and it is zero else.
Find:
(a) The value of c
(b) E(X)
(c) E(Y)
(d) E(X|Y)
(e) Var(X|Y)
(f) The MMSEE of eX given Y , E(eX|Y )
(g) Are X and Y independent?
fx,y(x, y) = c(2²/y)
1. Let X be an RV with density f(x) = ¼arosinx + c, x E [-1,11 (f(x) = 0 elsewhere). (a) Compute the constant c. (b) Compute the DF of X. (c) Compute the DF of the RV Y d) Compute P( <0.5) X2.
1. Let X be an RV with density f(x) = ¼arosinx + c, x E [-1,11 (f(x) = 0 elsewhere). (a) Compute the constant c. (b) Compute the DF of X. (c) Compute the DF of...