b. Let X(t) = X(u, t) denote a random process described byt≥0elsewhereX(t) =0,where, u is a realization of a uniform (0,1) random variable. DefineY(t) = Y(u, t) as follows:-utY(t)www.{Compute the correlation Ry(t₁, t₂).1, X(t) > e-²0, elsewhere.

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A First Course in Probability (10th Edition)

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Chapter1: Combinatorial Analysis

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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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E(XY) – E(X)E(Y) 1. The correlation of X and Y is defined as px,y if X and Y are independent, then px,y = 0. Quickly show that o(X)o(Y) 2. Let X be a random variable with P(X = k) = | for k = -1,0, 1. Let Y = X². (a) What is the p.m.f. of Y? (b) Are X and Y independent? (c) Calculate E(XY) – E(X)E(Y). (d) True or false (for any random variables X and Y): X and Y are independent if and only if E(XY) – E(X)E(Y). 3. An exam consists of twenty multiple choice questions. Each question has five different possible answers, only one of which is correct. A person randomly guesses on each question. What is the variance of the number of questions that are correct? 4. Suppose ten people get on an elevator. There are eight buttons representing floors on this elevator. Each person presses a random button. The elevator will only stop at a floor if at least one person pressed that floor's button. What is probability the elevator stops at exactly six floors? What is the variance of the number of floors…
17. Let die die = {(W₁ W₂) : w₁ = 1, 2, 3, 4, 5, 6; w₂= 1,2,3,4,5,6} and consider two random variables X(W₁=W₂) =W₁ + W₂= Y(w₁, W₂) = W₁ - W₂. Compute Cov(X, Y). Are they independent?
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The joint PDF of two jointly continuous random variables X and Y is S c(x2 + y?) for 0 < x < 1 and 0 < y < 1, fx,x(x, y) ‚Y otherwise. c = 3/2. E(Y) = 5/8. 3(2X2 + 1) 4(3X² + 1)' Show that E(Y|X) :
Consider a random process X(t) = A cos o t where a' is a constant and A is a random variable uniformly distributed over (0, 1). Find the auto correlation and covariance of X(t).
X is a continuous random variable. Note that X = max (0, X) – max (0, -X) show that E (X)= P (X > t) dt – | E P(X < t) dt
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7. For simple linear regression, we assume that Y = Bo + BIX +e, where e - N(0,0?) and X is fixed (not random). We collect n i.i.d, training sample (x),y1)....(XYn)). Prove that the (Bo.B1) estimated through minimizing RSS equals to the one through maximizing likelihood.
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Let X(t) = X(u, t) denote a random process described by t≥0 { 0, elsewhere X(t)= e Y(t) = -ut where, u is a realization of a uniform (0,1) random variable. Define Y(t) = Y(u, t) as follows: = { Compute the correlation Ry(t₁, t₂). 7 1, X(t) > e-2 0, elsewhere.