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MATH 255 - Probability and Statistics Final Exam Solutions 5 January 2025 Problem 1. [10pt] The joint pdf of random variables X and Y is given by: fX,Y (x, y) =    c if (x, y) ∈ S, 0 otherwise, where c is a constant and S is the set shown in the plot. (a) Find the least mean square (LMS) estimator g(X) of Y . E[Y |X] =    0.5 if X ∈ [0, 1] ∪ [2, 3] , 1 if X ∈ [1, 2]. Before we start solving the problem, let us first find the value of c. Since the joint...

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...

MATH-225 Final Exam — 20.05.2021 — Solutions 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...

MATH-225 Final Exam — Fall 2021 — Solutions Q 1. Let A be a 3 × 3-matrix, B a 3 × 2-matrix, and z a nonzero 3 × 1-matrix. Assume that: (i) B 1 0 0 0 6= B. (ii) B 0 0 0 1 6= B. (iii) B −2 0 0 1 = AB. (iv) det(A) = 2. (v) z is in Null(B T ). (vi) A = AT . Then: (a) (4 points) Compute det(A − 2I). (b) (4 points) Compute rank(B). (c) (4 points) Compute BTAz. (d) (4 points) Compute Az in terms of z. (e) (4 points) If B =   1 1 1 1 2 −1   then com...

Bilkent University Fall Math255 Probability and Statistics Midterm 2 Solutions Problem 1. [6 pts] Let (X, Y ) be uniformly distributed over the triangular region with corners at (0, 0), (1, 0), and (0, 1) in the x − y plane, i.e., fX,Y (x, y) = ( 2, x ≥ 0, y ≥ 0, x + y ≤ 1, 0, otherwise. (a) (3 pts) Compute the probability P 2X2 > Y
. (b) (3 pts) Compute P Z ≤ 3/4
where Z is defined as Z = max(X, Y ). (a) The probability in question is given by 2 (the density...

Bilkent University Fall Math255 Probability and Statistics Midterm 1 Solutions Problem 1. [6 pts] There are three coins each with possible outcomes heads (H) and tails (T). Coin 1 is a fair coin with equally likely outcomes. Coins 2 and 3 are bent coins, having probability of heads 5/6 and 1/6, respectively. Two of the three coins are picked at random and each flipped once. Let A be the event that coin 1 is one of the two coins picked. Let B be the event that the outcomes on the flipped ...

Bilkent University Fall 2020-21 Math 255 Introduction to Probability and Statistics Final Exam 7 Jan 2021 • Write your name, student number, and Math 255 section number on all your answer sheets. Number the sheets. • The exam consists of four problems. 50 points. • This is a closed book exam. One A4-size two-sided page of notes is allowed. No calculators. • You may receive no credit on correct answers if not fully justified. Simplify sums and integrals where possible to receive...

Bilkent University Spring 2020-21 Math 255 Introduction to Probability and Statistics Final Exam 28 May 2021 • Write your name, student number, and Math 255 section number on all your answer sheets. Number the sheets. • The exam consists of three problems. 30 points. • This is a closed book exam. One A4-size two-sided page of notes is allowed. No calculators. • You may receive no credit on correct answers if not fully justified. Simplify sums and integrals where possible to rec...

Bilkent University Spring 2020-21 Math 255 Introduction to Probability and Statistics Final Exam 28 May 2021 Solutions 1. [10 pts] Let X be a Bernoulli(θ) random variable where θ is modeled as a sample of a random variable Θ uniform on [0, 1]. In other words, we use a Bayesian model for the pair (Θ, X) so that fΘ(θ) = ( 1, 0 ≤ θ ≤ 1, 0, otherwise, and pX|Θ(x|θ) = ( θ, x = 1, 1 − θ, x = 0. (a) (5 pts) Compute the Least Mean Squares (LMS) estimate of Θ, namely Θ( ...

P1. (5 points) Let X and Y be two independent drawings from the uniform distribution on [0, a], with a > 0 a given constant. Let Z = X -Y| be the distance between the two points. Find the CDF Fz(z). Show your work by drawing a figure that explains how you calculate Fz(z). 2<0 27 22 Fz(z) = a 92 05259 279 Use this space to show your work for Problem 2 only. S1, if the ith dight drawn is Xi= (o, otheri 7 =) mean of 4th4xyt--+ Xpgon0 10008 (mgon ot X;) 10,00*= 1000 =EEY] ) vav...

That Sugar Film Movie Guide 1.What did the narrator give up eating when he met his girlfriend? 2. What percent of items on the shelves of the grocery store do not contain sugar? 3. How many pounds of sugar do Americans eat a year? 4. How many teaspoons of sugar do Australians eat a day? 5. What is Glucose used for? 6. How must the narrator eat the