Questions and Answers of Introduction To Probability Statistics

Suppose X ∼ Uniform(1, 2), and given X = x, Y is exponential with parameter λ = 1/x.a. Find the linear MMSE estimate of X given Y .b. Find the MSE of this estimator.c. Check that E[X̂Y ] = 0.
Let X be an unobserved random variable with EX = 0, Var(X) = 5. Assume that we have observed Y1, Y2, and Y3 given bywhere EW1 = EW2 = EW3 = 0, Var(W1) = 2, Var(W2) = 5, and Var(W3) = 3. Assume that
Let X be an unobserved random variable with EX = 0, Var(X) = 4. Assume that we have observed Y1 and Y2 given byY1 = X +W1,Y2 = X +W2,where EW1 = EW2 = 0, Var(W1) = 1, and Var(W2) = 4. Assume that W1,
Consider two random variables X and Y with the joint PMF given by the table below.a. Find the linear MMSE estimator of X given Y , (X̂L).b. Find the MMSE estimator of X given Y , (X̂M).c. Find the
Suppose that t he random variable X is transmitted over a communication channel. Assume that the received signal is given by Y = X +W, where W ∼ N(0,σ2) is independent of X. Suppose that X = 1
Find the average error probability in Example 9.10Example 9.10Suppose that t he random variable X is transmitted over a communication channel. Assume that the received signal is given by Y = X +W,
Consider two random variables X and Y with the joint PMF given by the table below.a. Find the linear MMSE estimator of X given Y , ( X̂L).b. Find the MSE of X̂L.c. Find the MMSE estimator of X
Explain why the following approximations hold: a. pjj (8) b. pkj (6) 1+9jjd, for all j = S. 8gkj, for k ‡ j.
Consider the Markov chain shown in Figure 11.36. Assume that 0 < p < q. Does this chain have a limiting distribution? For all i, j ∈ {0, 1, 2,⋯}, find
Consider the co ntinuous Markov chain of Example 11.17: A chain with two states S = {0, 1} and λ0 = λ1 = λ > 0. In that example, we found that the transition matrix for any t ≥ 0 is given
Consider the Markov chain shown in Figure 11.37. Assume that p > q > 0. Does this chain have a limiting distribution? For all i, j ∈ {0, 1, 2,⋯}, find q+r P 9 lim P(Xn=jXo =
The generator matrix for the continuous Markov chain of Example 11.17 is given byFind the stationary distribution for this chain by solving πG = 0.Example 11.17Consider a continuous Markov chain
Two gamblers, call them Gambler A and Gambler B, play repeatedly. In each round, A wins 1 dollar with probability p or loses 1 dollar with probability q = 1 −p (thus, equivalently, in each round B
Let W(t) be a standard Brownian motion. For all s, t ∈ [0,∞), find CW(s, t) = Cov(W(s),W(t)).
Let N = 4 and i = 2 in the gambler's ruin problem. Find the expected number of rounds the gamblers play until one of them wins the game.
Let W(t) be a standard Brownian motion.a. Find P(1 < W(1) < 2).b. Find P(W(2) < 3|W(1) = 1).
The Poisson process is a continuous-time Markov chain. Specifically, let N(t) be a Poisson process with rate λ.a. Draw the state transition diagram of the corresponding jump chain.b. What are the
Consider a continuous-time Markov chain X(t) that has the jump chain shown in Figure 11.38. Assume λ1 = λ2 = λ3, and λ4 = 2λ1. H 1 4 1 2 3 1 Figure 11.38 - The jump chain for the Markov chain of
Consider a continuous-time Markov chain X(t) that has the jump chain shown in Figure 11.39. Assume λ1 = 1, λ2 = 2, and λ3 = 4.a. Find the generator matrix for this chain.b. Find the limiting
Consider the queuing system of Problem 3 in the Solved Problems Section. Specifically, in that problem we found the following generator matrix and transition rate diagram:The transition rate diagram
Let W(t) be the standard Brownian motion.a. Find P(−1 < W(1) < 1).b. Find P(1 < W(2) +W(3) < 2).c. Find P(W(1) > 2|W(2) = 1).
Let W(t) be a standard Brownian motion. FindP(0 < W(1) +W(2) < 2, 3W(1) −2W(2) > 0).
Let W(t) be a standard Brownian motion. DefineNote that X(0) = X(1) = 0. Find Cov(X(s),X(t)), for 0 ≤ s ≤ t ≤ 1. X(t) = W(t) - tW (1), for all t € [0,00).
Let W(t) be a standard Brownian motion. Let a > 0. Define Ta as the first time that W(t) = a. That isa. Show that for any t ≥ 0, we haveb. Using Part (a), show thatc. Using Part (b), show that
Let W(t) and U(t) be two independent standard Brownian motions. Let −1 ≤ ρ ≤ 1. Define the random process X(t) asa. Show that X(t) is a standard Brownian motion.b. Find the covariance and
Simulate tossing a coin with probability of heads p.
Write codes to simulate tossing a fair coin to see how the law of large numbers works.
Generate a Binomial(50; 0:2) random variable.
Give an algorithm to simulate the value of a random variable X such that P(X= 1) = 0.35 P(X2) = 0.15 P(X= 3) = 0.4 P(X= 4) = 0.1
Generate an Exponential(1) random variable.
Generate a Gamma(20,1) random variable.
Generate a Poisson random variable. In this example, use the fact that the number of events in the interval [0; t] has Poisson distribution when the elapsed times between the events are Exponential.
Generate 5000 pairs of normal random variables and plot both histograms.
Let A,B, and C be three events with probabilities given below:a. Find P(A|B).b. Find P(C|B).c. Find P(B|A ∪ C).d. Find P(B|A,C) = P(B|A ∩ C). В 0.1 (0.05 C 0.1 0.1 0.15 A 0.2 0.1 S
For each of the following Venn diagrams, write the set denoted by the shaded area.a.b.c.d. A B S
Suppose th at the universal set S is defined as S = {1, 2,⋯, 10} and A = {1, 2, 3}, B = {X ∈ S : 2 ≤ X ≤ 7}, and C = {7, 8, 9, 10}.a. Find A ∪ B.b. Find (A ∪ C) −B.c. Find Ā ∪ (B
The following sets are used in this book:The set of natural numbers, N = {1, 2, 3,⋯}.The set of integers, Z = {⋯,−3,−2,−1, 0, 1, 2, 3,⋯}.The set of rational numbers Q.The set of real
When working with real numbers, our universal set is R. Find each of the following sets.a. [6, 8] ∪ [2, 7)b. [6, 8] ∩ [2, 7)c. [0, 1]cd. [6, 8]−(2, 7)
Here are som e examples of sets defined by stating the properties satisfied by the elements:If the set C is defined as C = {x|x ∈ Z,−2 ≤ x < 10}, then C = {−2,−1, 0,⋯, 9}.If the set D is
Here are some examples of sets and their subsets:If E = {1, 4} and C = {1, 4, 9}, then E ⊂ C.N ⊂ Z.Q ⊂ R.
A coin is tossed twice. Let S be the set of all possible pairs that can be observed, i.e., S = {H,T} ×{H,T} = {(H,H), (H,T), (T,H), (T,T)}. Write the following sets by listing their elements.a. A:
Let A = {1, 2,⋯, 100}. For any i ∈ N, Define Ai as the set of numbers in A that are divisible by i. For example:a. Find |A2|,|A3|,|A4|,|A5|.b. Find |A2 ∪ A3 ∪ A5|. A₂ = {2,4,6,, 100}, As
If the universa l set is given by S = {1, 2, 3, 4, 5, 6}, and A = {1, 2}, B = {2, 4, 5},C = {1, 5, 6} are three sets, find the following sets:a. A ∪ Bb. A ∩ Bc. Ād. B̄e. Check De Morgan's law
Suppose th at A1, A2, A3 form a partition of the universal set S. Let B be an arbitrary set. Assume that we knowFind |B|. BnA₁ BnA₂ BnA₂ = 10, = 20, = 15.
In a party,There are 10 people with white shirts and 8 people with red shirts;4 people have black shoes and white shirts;3 people have black shoes and red shirts; The total number of people with
Determine whether each of the following sets is countable or uncountable. a. A = {1,2,, 10¹0} b. B = {a+b√2 a, b = Q} C. C = {(X,Y) = R²| x² + y² ≤ 1}.
Consider the function f : R → R, defined as f(x) = x2. This function takes any real number x and outputs x2. For example, f(2) = 4.Consider the function g : {H,T} → {0, 1}, defined as g(H) = 0
LetDefine Find A. n-1 An A₁ = [0, ¹=¹) = {x € R| 0
We toss a coin three times and observe the sequence of heads/tails. The sample space here may be defined asS = {(H,H,H), (H,H,T), (H,T,H), (T,H,H), (H,T,T), (T,H,T), (T,T,H), (T,T,T)}.
LetDefine Find A. An = [0, 1) = { x = R| 0
In a presidential election, there are four candidates. Call them A, B, C, and D. Based on our polling analysis, we estimate that A has a 20 percent chance of winning the election, while B has a 40
In this problem our goal is to show that sets that are not in the form of intervals may also be uncountable. In particular, consider the set A defined as the set of all subsets of N:We usually denote
You roll a fair die. What is the probability of E = {1, 5}?
Using the axioms of probability, prove the following:a. For any event A, P(Ac) = 1 −P(A).b. The probability of the empty set is zero, i.e., P(∅) = 0.c. For any event A, P(A) ≤ 1.d. P(A −B) =
Suppose we have the following information:1. There is a 60 percent chance that it will rain today.2. There is a 50 percent chance that it will rain tomorrow.3. There is a 30 percent chance that it
Two teams A and B play a soccer match, and we are interested in the winner. The sample space can be defined aswhere a shows the outcome that A wins, b shows the outcome that B wins, and d shows the
I play a gambli ng game in which I will win k −2 dollars with probability 1/2k for any k ∈ N , that is,With probability 1/2, I lose 1 dollar;With probability 1/4, I win 0 dollar;With probability
Let A and B be two events such thata. Find P(A ∩ B).b. Find P(Ac ∩ B).c. Find P(A −B).d. Find P(Ac −B).e. Find P(Ac ∪ B).f. Find P(A ∩ (B ∪ Ac)). P(A) = 0.4, P(B) = 0.7,P(AUB) = 0.9
I roll a fair die twice and obtain two numbers: X1 = result of the first roll, and X2 = result of the second roll. Write down the sample space S, and assuming that all outcomes are equally likely
Your friend tell s you that she will stop by your house sometime after or equal to 1 p.m. and before 2 p.m., but she cannot give you any more information as her schedule is quite hectic. Your friend
Consider a random experiment with a sample spaceSuppose that we know:where c is a constant number.a. Find c.b. Find P({2, 4, 6}).c. Find P({3, 4, 5,⋯}). S = {1,2,3,}.
I roll a fair die twice and obtain two numbers: X1 = result of the first roll, X2 = result of the second roll.a. Find the probability that X2 = 4.b. Find the probability that X1 + X2 = 7.c. Find the
I roll a fair die. Let A be the event that the outcome is an odd number, i.e., A = {1, 3, 5}. Also let B be the event that the outcome is less than or equal to 3, i.e., B = {1, 2, 3}. What is the
For three events, A, B, and C, with P(C) > 0, we haveP(Ac|C) = 1 −P(A|C);P(∅|C) = 0;P(A|C) ≤ 1;P(A −B|C) = P(A|C) −P(A ∩ B|C);P(A ∪ B|C) = P(A|C) +P(B|C) −P(A ∩ B|C);if A ⊂ B then
I roll a fair die twice and obtain two numbers X1 = result of the first roll and X2 = result of the second roll. Given that I know X1 + X2 = 7, what is the probability that X1 = 4 or X2 = 4?
Four teams A,B,C, and D compete in a tournament, and exactly one of them will win the tournament. Teams A and B have the same chance of winning the tournament. Team C is twice as likely to win the
Let T be the time needed to complete a job at a certain factory. By using the historical data, we know thata. Find the probability that the job is completed in less than one hour, i.e., find P(T ≤
a. Let A1,A2,A3,⋯ be a sequence of increasing events, that isShow thatb. Using part(a), show that if A1,A2,⋯ is a decreasing sequence of events, i.e., A₁ CA2 C A3 C***
Consider a family that has two children. We are interested in the children's genders. Our sample space is S = {(G,G), (G,B), (B,G), (B,B)}. Also assume that all four possible outcomes are equally
You choose a point (A,B) uniformly at random in the unit square {(x, y) : x, y ∈ [0, 1]}.What is the probability that the equationhas real solutions? 1 B 0 (A, B) A 1 X
In a factory there are 100 units of a certain product, 5 of which are defective. We pick three units from the 100 units at random. What is the probability that none of them are defective?
I pick a random number from {1, 2, 3,⋯, 10}, and call it N. Suppose that all outcomes are equally likely. Let A be the event that N is less than 7, and let B be the event that N is an even number.
For any sequence of events A1,A2,A3,⋯, prove P(ŨA.) - Im P (U₁.). Р A₁ lim (vū) P(₁) - I'm P(₁.) Р = lim =1
I toss a coin repeatedly until I observe the first tails at which point I stop. Let X be the total number of coin tosses. Find P(X = 5).
Suppose that the probability of being killed in a single flight is pc = 1/4×106 based on available statistics. Assume that different flights are independent. If a businessman takes 20 flights per
Suppose that, of all the customers at a coffee shop,70% purchase a cup of coffee;40% purchase a piece of cake;20% purchase both a cup of coffee and a piece of cake.Given that a randomly chosen
Two basketball players play a game in which the y alternately shoot a basketball at a hoop. The first one to make a basket wins the game. On each shot, Player 1 (the one who shoots first) has
I have three bags that each contain 100 marbles:Bag 1 has 75 red and 25 blue marbles;Bag 2 has 60 red and 40 blue marbles;Bag 3 has 45 red and 55 blue marbles.I choose one of the bags at random and
A real number X is selected uniformly at random in the continuous interval [0, 10]. (For example, X could be 3.87.)a. Find P (2 ≤ X ≤ 5).b. Find P (X ≤ 2|X ≤ 5).c. Find P (3 ≤ X ≤ 8|X ≥
In Example 1.24, suppose we observe that the chosen marble is red. What is the probability that Bag 1 was chosen?Example 1.24:I have three bags that each contain 100 marbles:Bag 1 has 75 red and 25
A professor thinks students who live on campus are more likely to get As in the probability course. To check this theory, the professor combines the data from the past few years:a. 600 students have
Recall thatConsider the following functiondefined asFor example,a. Determine the domain and co-domain for f.b. Find range of f:Range(f).c. If we know f(x) = 2, what can we say about x? {H,T}³ =
Let X be a random variable with the following CDF:a. What kind of random variable is X (discrete, continuous, or mixed)?b. Find the (generalized) PDF of X.c. Find P(X > 0.5), both using the CDF and
A company makes a certain device. We are interested in the lifetime of the device. It is estimated that around 2% of the devices are defective from the start so they have a lifetime of 0 years. If a
A continuous random variable is said to have a Laplace(μ, b) distribution [14] if its PDF is given by
Let X ∼ Laplace(0, b), i.e.,where b > 0. Define Y = |X|. Show that Y ∼ Exponential (1/b). fx(æ) 1 ਆ) (E), exp 26 b
A continuous random variable is said to have the standard Cauchy distribution if its PDF is given byIf X has a standard Cauchy distribution, show that EX is not well-defined. Also, show EX2 = ∞.
A continuous random variable is said to have a Rayleigh distribution with parameter σ if its PDF is given bywhere σ > 0.a. If X ∼ Rayleigh(σ), find EX.b. If X ∼ Rayleigh(σ), find the CDF of
A continuous random variable is said to have a Pareto(xm,α) distribution [15] if its PDF is given bywhere xm,α > 0. Let X ∼ Pareto(xm,α).a. Find the CDF of X, FX(x).b. Find P(X > 3xm|X > 2xm).c.
Let Z ∼ N(0, 1). If we define X = eσZ+μ, then we say that X has a log-normal distribution with parameters μ and σ, and we write X ∼ LogNormal(μ, σ).a. If X ∼ LogNormal(μ,σ), find the
Let X1, X2, ⋯, Xn be independent random variables with Xi ∼ Exponential(λ). Define Y = X1 +X2 +⋯+Xn.As we will see later, Y has a Gamma distribution with parameters n and λ, i.e., Y ∼
Let X and Y be as defined in Problem 1. I define a new random variable Z = X −2Y.a. Find the PMF of Z.b. Find P(X = 2|Z = 0).Problem 1Consider two random variables X and Y with joint PMF given in
A box contains two coins: a regular coin and a biased coin with P(H) = 2/3. I choose a coin at random and toss it once. I define the random variable X as a Bernoulli random variable associated with
Consider the set of points in the grid shown in Figure 5.4. These are the points in set G defined asSuppose that we pick a point (X, Y) from this grid completely at random. Thus, each point has a
Consider two random variables X and Y with joint PMF given bya. Show that X and Y are independent and find the marginal PMFs of X and Y.b. Find P(X2 + Y2 ≤ 10). PXY (k, l) = 1 , for k, l = 1,2,3,
Let X and Y be as defined in Problem 1. Also, suppose that we are given that Y = 1.a. Find the conditional PMF of X given Y = 1. That is, find PX|Y (x|1).b. Find E[X|Y = 1].c. Find V ar(X|Y =
Let X ∼ Geometric(p). Find EX by conditioning on the result of the first "coin toss."
The number of customers visiting a store in one hour has a Poisson distribution with mean λ = 10. Each customer is a female with probability p = 3/4 independent of other customers. Let X be the
Suppose that the number of customers visiting a fast food restaurant in a given day is N ∼ Poisson(λ). Assume that each customer purchases a drink with probability p, independently from other
Let X ∼ Geometric(p). Find Var(X) as follows: Find EX and EX2 by conditioning on the result of the first "coin toss", and use Var(X) = EX2 −(EX)2.

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