Given a marginal p.m.f of x: f₁(x) = and a marginal p.m.f of y: f₂(y) = 2 X & Y are stochastically independent r.v.³ then the J.p.m.f of x and y is " f(x, y) = › = {1/ f(x,y)= f(x, y) = 0 > . for x = 2,8 O.W , for y = 1,3 O.W for x = 2,8 y = 1,3 O.W Option 3 for x = 2,8 y = 1,3 O.W Option 2 O for x = 2,8 y = 1,3 O.W
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![Given a marginal p.m.f of x: f₁(x) =
0
1
and a marginal p.m.f of y: f₂(y) = 2
X & Y are stochastically independent r.v.³
then the J.p.m.f of x and y is
»= {1
f(x, y) =
f(x, y) =
f(x,y)=4
0
, for x = 2,8
O.W
, for y = 1,3
O.W
for x = 2,8
y = 1,3
O.W
Option 3
for x= 2,8
y = 1,3
O.W
Option 2
for x = 2,8
y = 1,3
O.W](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5c1ce3ba-3c5c-4193-afeb-11d393ea9a8c%2Fa69b7271-1c0e-412c-9cfb-177478c87e1d%2F19p0jne_processed.jpeg&w=3840&q=75)
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- Consider randomly selecting n segments of pipe and determining the corrosion loss (mm) in the wall thickness for each one. Denote these corrosion losses by Y₁' Yn. The article "A Probabilistic Model for a Gas Explosion Due to Leakages in the Grey Cast Iron Gas Mains"+ proposes a linear corrosion model: Y; = t;R, where t; is the age of the pipe and R, the corrosion rate, is exponentially distributed with parameter 1. Obtain the maximum likelihood estimator of the exponential parameter (the resulting mle appears in the cited article). [Hint: If c> 0 and X has an exponential distribution, so does cX.] O Â = O Â = O λ = * * O n Srit) i = 1 n j = 1 Σ i = 1 n n i = 1 n LY i = 1 n 0 1 = L (rt) 2 i = 1 | = 1 n n Y; | = 12. Suppose that Y1 ~ Exp(3) and Y2 ~ Exp(2) are independent. (a) Determine the joint pdf of Y1 and Y2. (b) Use the Jacobian method to determine the pdf of U = Y1 + Y2. (c) Use the mgf method to determine the distribution of V = 4Y1 + 6Y2. The distribution of V will be one of the "famous" ones in your Super Handout.5. Random variables X and Y are independent and identically distributed. The moment generating function of X+Y is: mx+y(t) = e6t+9t*, ,te R. X+Y %3D Calculate P(X > 37.92).
- 2. We would like to fit a linear regression estimate to the dataset {(x®,y@),(x), y),., (x(N), g/N)} with x e RM by minimizing the ordinary least square (OLS) objective function: N M -Συ, .(i) J(w): j=1 Specifically, we solve for each coefficient wk (1< k < M) by deriving an expression of Wk from the critical point J(w) the dataset (x(1), y(1)), (x(2), y(2)), . 0. What is the expression for each wk in terms of … , (x(^), y(N) and w1, , wk-1, Wk+1; *** , WM? .. .. Select one: E, (y() –D,-1,j+k W;x;") i=D1 Wk = =1 O WkFall23 HW3.3. Let y0nd. Exp(0.), Inv-Gamma (a, 1), for i=1,..., n. Find the empirical Bayes estimator of 0,, i 1,..., n (Note: If z~ Exp(0), then p(x)=1/0e-1/0 and if Inv-Gamma(a, 3), then p(x) = 1/{3ªT(a)}r-a-¹e-¹/³).).Let Y1, Y2, ..., Yn denote a random sample of size n from a population whose density is given by 3y2 0 < y< 0, f (y|0) = өз 0, elsewhere. 1.1. Show that Y(n) max(Y1, Y2, ... , Yn) is sufficient for 0.Consider the Keynesian consumption function Yt = B₁ + B₂x2t + &t where yt is per capita consumption, and x2+ is per capita income. The coefficient ₂ is interpreted causally as the marginal propensity to consume, and we expect 0Q1: Let X have p.d.f f(x) : =7,x = 1,2, 3 and X1 and X2 are stochastically 6. independent. Let Y1 = X,X2 and let Y2 = X2. Find g(y1, y2)and g,(y1). ***********************: ************: ******************Given a J. p. d. f. of x and y: f(x,y) = {e= (x + y) - x and y are stochastically independent. True O False for x > 0, y > 0 0.w2. Assuming the dichotomous disease model, with Y1 and Y2 being two relatives with relatedness R, show that (a) Cov(Y1, Y2) = P(Y1 = Y2 = 1) – K². (b) express R as a function of the covariance and K.8. (a) Assume process X, satisfies X,-aX-1+ Z where Z, is WN(0, a?) and Ja< 1. Use the Yule-Walker equation to find the best linear predictor P(X+iXn Xn-1). Find the mean-square error of the predictor. = Z, + aZ,-1 + bZ,-2, where Z, is WN(0, a). Use the (b) Assume process X, satisfies X, Durbin-Levinson algorithm to find the linear predictor P(X+i|X Xn-1)-Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON