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MATH 255 - Probability and Statistics Midterm Exam II Solutions 24 November 2024 Problem 1. [20pt] The joint pdf of random variables X and Y is given by: fX,Y (u, v) =    8uv u2 + v 2 ≤ 1, u ≥ 0, v ≥ 0 0 else (a) Find the marginal pdf fX(x). Verify your answer by showing that fX(x) is a valid pdf. fX(x) =    4x(1 − x 2 ) 0 ≤ x ≤ 1 0 else The marginal pdf of X can be written as: fX(x) = Z ∞ −∞ fX,Y (x, y)dy = Z √ 1−x2 0 8xydy = 8...

MATH 255 - Probability and Statistics Solutions to Midterm Exam I Problem 1. [6pt] Suppose A, B, and C are events for a probability experiment such that A and B are mutually independent, P(A) = P(B) = P(C) = 0.5, P(A ∩ C) = P(B ∩ C) = 0.3, and P(A ∩ B ∩ C) = 0.1. Fill in the probabilities of all events in the Karnaugh map below. Show your work. Due to mutual independence, we have P(A ∩ B) = 0.25. P(A ∩ B ∩ C) = 0.1 P(Ac ∩ B ∩ C) = 0.2 (+0.5 pt) P(A ∩ Bc ∩ C) = 0.2 ...

Bilkent University Spring 2020-21 Math 255 Probability and Statistics Midterm 1, March 8, 2021 Solutions 1. [6 pts] Let A, B, C be three independent events in a probability space (Ω, P) with P(A) = 0.2, P(B) = 0.3, and P(C) = 0.4. Compute the following probabilities. Each part is 2 pts. (a) P((A ∪ B) ∩ C c ) (b) P(Bc ∪ C c | Ac ) (c) P(A ∩ B | B ∩ C) Solution. (a) The main point of this part is to observe that if A, B, C are independent events then A ∪ B and C c ...

Matn 255 5?““3 2014 -20 M.‘Jl—&rm 1~ AVSWUS All numeric answers must be simplified to a real number or a fraction of two integers with no common factors. Show your work legibly to maximize partial credit. Problem 1. [6 pts] Consider a probability space (@, P) and let 4, B,C C Q be three events with P(A)=0.1, P(B) = 0.2, and P(C) = 0.3. (a) Compute P[(AUB)C|B*UC] under the assumption that A, B, C are independent. (b) Assume instead that (i) A, B, and C are pairwise independen...

DO NOT USE THIS PAGE FOR SOLUTIONS. WRITE ONLY YOUR FINAL ANSWERS ON THIS PAGE. Problem 1. [10 pts] You roll a fair six-sided die, and then you flip a fair coin the number of times shown by the die. Let X denote the number of heads obtained. Find E[X] and var(X). Numerical answers are required. Show your work in detail. B/X)= 7/4 Va (X) = 77/48 Problem 2. [10 pts] We obtain an observation X=+Z of a parameter and wish to estimate using a Bayesian approach. Suppose that is a discrete rand...

MATH 255 - Probability and Statistics Final Exam Solutions Problem 1. [8pt] Let X1, X2, . . . be independent random variables that are uniformly distributed over [0, 1]. Show that the sequence of Y1, Y2, . . . converges with probability 1 to some limit and identify the limit, for the case where Yn is the sampled geometric mean, given by Yn = Yn i=1 Xi !1/n Solution: limn→∞ Yn i=1 Xi !1/n = limn→∞ exp  log Yn i=1 Xi !1/n  = limn→∞ exp 1 n log Yn ...

MATH 255 - Probability and Statistics Final Exam Solutions Problem 1) Let fX|Θ(x|θ) = ( θe−θx if x ≥ 0 0 if x < 0 and fΘ(θ) = ( αe−αθ if θ ≥ 0 0 if θ < 0 Find the MAP and LMS estimates of θ for a single observation X = x. Hint: For exponential random variables, we have R ∞ 0 λe−λxdx = 1 and R ∞ 0 x 2λe−λxdx = 2 λ2 . We can first find the posterior distribution (2pt): fΘ|X(θ|x) = fX|Θ(x|θ)fΘ(θ) fX(x) = αθe−(α+x)θ R ∞ 0 ...

MATH 255 - Probability and Statistics Midterm Exam II Solutions Problem 1) Suppose that X and Y have the joint PDF: fX,Y (x, y) = ( e −x if 0 ≤ y ≤ x 0 o.w. Find the marginal PDF of X and the conditional PDF of Y given X. (a) The marginal PDF of X is given by fX(x) = Z ∞ −∞ fX,Y (x, y)dy = ( R x 0 e −xdy x ≥ 0 0 o.w. = ( xe−x x ≥ 0 0 o.w. (b) The conditional PDF of Y given X is undefined if x ≤ 0 . For x > 0, it can be written as fY |X(y|x) = ( e...

MATH 255 - Probability and Statistics Solutions to Midterm Exam II Problem 1. [10pt] A stick is broken into three pieces by picking two points independently and uniformly along the stick, and breaking the stick at those two points. What is the probability that the three pieces can be assembled into a triangle? Solution: Consider the case where x > y without loss of generality. Then, the segments have lengths: y, x − y, and 1 − x. To form a triangle, they must satisfy the triangle in...

MATH-225 Final Exam — 20.05.2021 — 13:00–16:00 N.B. Correct answers without sufficient correct mathematical explanations will not get full credit. Q 1: Let A be an n×n matrix, λ1 an eigenvalue of A, and let In denote the identity matrix of size n×n. Recall that the multiplicity of λ1 is the largest integer k such that (λ − λ1) k is a factor of the characteristic polynomial |λIn − A|. (a) (5 pts) Show by an example that the dimension of Null(λ1In − A) can be different f...