Problems4.3. Suppose that X and Y are jointly discrete random variables withfor x = 1, 2, .n and2p(x, y) = {n(n + 1)y = 1, 2, .otherwiseCompute Px(x) and py(y) and determine whether X and Y are independent.

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Answered: Problems 4.3. Suppose that X and Y are…

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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a) Let X₁, X₂, X3,..., Xn be a random sample of size n from population X. Suppose that X~N(0, 1) and Y = – Σ₁=1 Xi √n - en. What is the probability that Y² is between 0.02 and 5.02?
Suppose we have three independent random variables X, Y and Z where... Var(X + 2Y) = 13, Var(2Y + 4Z) = 40, and Var(Z) = 2. a) If E(Y) = E(Z) = 1, what is E(Y2+Z2)? b) What is SD(X + Y + Z)?
IV) When a current X amperes flows through a resistance Y ohms, the power generated is given by w=x²y watts. Suppose the current and resistance are independent random variables with densities b = f(x)=6x (1-x), 0≤x≤1 X If the CDF of Wis Fw (w) = wª +6w - 8w 3/2, b ≤w ≤c, where a, b and care constants then a = C= fy(y)=2y, 0≤x≤1
5. Let {S, n>0} be a simple random walk with So = 0 and S₁ = X₁ + ... + X, for n ≥ 1, where X₁, i = 1,2,... are independent random variables with P(X; = 1) = p, P(X; = -1) = q = 1 -p for i>1. Assume p‡q. Put Fo= {2,0}, Fn = 0(X₁, X2, ..., Xn), n ≥ 1. Let b, a be two fixed positive integers. Define T = min{n: S₁ = -a or S₁₁=b}. Assume P(T<∞) = 1. Explain why one can use Doob's Optional Stopping Theorem to conclude that E[ZT] = 1. We can use the conclusion that: T is a stopping time with respect to the o-fields Fn, n ≥ 0. A) B) Define Z₁ = (q/p) Sn, n ≥ 0. {Zn, n ≥ 0} is a martingale with respect to the o-fields Fn, n ≥ 0.
5. Suppose that X,, X2, -, Xm and Y1,Y2, ,Y, are independent random samples from N(Hx,o) and N(Hy, o), respectively. a. Find E(X – Ý). b. Find V(X – Ý)
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13-17. If a random variable X assumes the values 0 and 1 with P(X=0) = 3 P(X= 1), then nX) is : () 16 (ii) is 3 (iii) 16 (iv) 4. i6 m n t m 2/6
2. Let X₁, X2, X3,..., be independent identically distributed Unif(0, 1) random variables. Let Sn be the largest gap between any two of the points {X1, X2,..., Xn} (e.g. if n = 3 and (X₁, X2, X3) = (0.1, 0.5, 0.7) then S3 = 0.4). Show that for any x > 0 lim P(nSn > x) = 1. n→∞ Hint: to lower bound nSn > x, if suffices to demonstrate that there is an empty interval of some length x/n which contains none of the points. Make a list of some candidate intervals and then use Cheby- shev's inequality to show one of them is empty with probability going to 1.
4. A particle moves on the x- axis starting at x = 0. With equal probability, the particle moves to the left or right one unit. Let X denote the position on the x- axis after four moves. (a) Find px(x). Notice that X can take on -4,-2,0, 2, 4. Fill out the table below. xPx(x) (b) What is the expected position of the particle after four moves. That is, compute E(X). Express your answer as a fraction in lowest terms.
24. Gaussian random variables X, and X,, for which, X, = 2, o = 9, X, =-1, ox, = 4 and Cxx, =-3, are transformed to new random variables Y, and Y, according to जापर डेरें| Y, =-X, + X, Y, =-2X,- 3X, (iii) Cy,Cy, .. Find
Directions: Solve the given problem. Show your solutions. Of all the registered automobiles in Colorado, 8% fail the state emissions test. Twelve automobiles are elected at random to undergo an emissions test. a. Find the probability that exactly four of them fail the test. b. Find the probability that fewer than four of them fail the test. c. Find the probability that more than three of them fail the test.
7. Suppose that the joint p.d.f. of two random variables X and Y is as follows: (4 – 2x – y) for x > 0, y > 0, f(x, y) = and 2x + y 2|X = 0.5).

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Problems 4.3. Suppose that X and Y are jointly discrete random variables with for x = 1, 2, .. y = 1, 2, otherwise n and 2 p(x, y) = {n(n + 1) Compute Px(x) and py(y) and determine whether X and Y are independent.