Questions and Answers of Introduction To Probability Statistics

In some applications, we need to work with complexvalued random processes. More specifically, a complex random process X(t) can be written as X(t) = Xr(t) +jXi(t), where Xr(t) and Xi(t) are two
Let {X(t), t ∈ R} be a continuous-time random process. The time average mean of X(t) is defined as (assuming that the limit exists in mean-square sense)Consider the random process {X(t), t ∈ R}
Let X(t) be a zero-mean Gaussian random process with RX(τ) = 8 sinc(4τ). Suppose that X(t) is input to an LTI system with transfer functionIf Y (t) is the output, find P(Y (2) H(f) = 2 0 |f|
Let X(t) be a white Gaussian noise process that is input to an LTI system with transfer functionIf Y (t) is the output, find P(Y (1) 0). |H(f)| = 2 0 1
Let X(t) be a WSS process. We say that X(t) is mean ergodic if 〈X(t)〉 (defined above) is equal to μX. Let A0, A1, A−1, A2, A−2, ⋯ be a sequence of i.i.d. random variables with mean EAi =
Let {X(t), t ∈ R} be a WSS random process. Show that for any α > 0, we have P(|X(t + r) - X(t)| > a) ≤ 2Rx(0) -2RX(T) a²
Let {X(t), t ∈ R} be a WSS random process. Suppose that RX(τ) = RX(0) for some τ > 0. Show that, for any t, we have X(t+r) = X(t), with probability one.
Let X(t) be a real-valued WSS random process with autocorrelation function RX(τ). Show that the Power Spectral Density (PSD) of X(t) is given by ∞ Sx(f) = Rx (7) cos(2n fr) dr. -∞
Let X(t) be a WSS process with autocorrelation functionAssume that X(t) is input to a low-pass filter with frequency responseLet Y (t) be the output.a. Find SX(f).b. Find SXY(f).c. Find SY (f).d.
Let X(t) and Y (t) be real-valued jointly WSS random processes. Show that SY X(f) = S∗XY (f), where, ∗ shows the complex conjugate.
Let X(t) be a WSS process with autocorrelation functionAssume that X(t) is input to an LTI system with impulse responseLet Y (t) be the output.a. Find SX(f).b. Find SXY (f).c. Find RXY (τ).d. Find
Let X(t) be a zero-mean WSS Gaussian random process with RX(τ) = e−πτ2. Suppose that X(t) is input to an LTI system with transfer functionLet Y (t) be the output.a. Find μY .b. Find RY (τ) and
Let X(t) be a white Gaussian noise with SX(f) = N0/2. Assume that X(t) is input to a bandpass filter with frequency responseLet Y (t) be the output.a. Find SY (f).b. Find RY (τ).c. Find E[Y (t)2].
The number of customers arriving at a grocery store can be modeled by a Poisson process with intensity λ = 10 customers per hour.1. Find the probability that there are 2 customers between 10:00 and
The number of orders arriving at a service facility can be modeled by a Poisson process with intensity λ = 10 orders per hour.a. Find the probability that there are no orders between 10:30 and 11.b.
Let N(t) be a Poisson process with intensity λ = 2, and let X1, X2, ⋯ be the corresponding interarrival times.a. Find the probability that the first arrival occurs after t = 0.5, i.e., P(X1 >
In this problem, our goal is to complete the proof of the equivalence of the first and the second definitions of the Poisson process. More specifically, suppose that the counting process {N(t), t ∈
Let X ∼ Poisson(μ1) and Y ∼ Poisson(μ2) be two independent random variables. Define Z = X +Y . Show that X|Z = n~ Binomial (n, μ1 f1 + f₂
Let {N(t), t ∈ [0,∞)} be a Poisson process with rate λ. Find the probability that there are two arrivals in (0, 2] or three arrivals in (4, 7].
Consider the Markov chain shown in Figure 11.7.a. Find P(X4 = 3|X3 = 2).b. Find P(X3 = 1|X2 = 1).c. If we know P(X0 = 1) = 1/3, find P(X0 = 1,X1 = 2).d. If we know P(X0 = 1) = 1/3, find P(X0 = 1,X1 =
Let N1(t) and N2(t) be two independent Poisson processes with rate λ1 and λ2 respectively. Let N(t) = N1(t) +N2(t) be the merged process. Show that givenWe can interpret this result as follows: Any
Consider a system that can be in one of two possible states, S = {0, 1}. In particular, suppose that the transition matrix is given bySuppose that the system is in state 0 at time n = 0, i.e., X0 =
Let {N(t), t ∈ [0,∞)} be a Poisson process with rate λ. Let T1, T2, ⋯ be the arrival times for this process. Show thatOne way to show the above result is to show that for sufficiently small
Consider the Markov chain shown in Figure 11.9. It is assumed that when there is an arrow from state i to state j, then pij > 0. Find the equivalence classes for this Markov chain. (2 (3) 8 Figure
Show that in a finite Markov chain, there is at least one recurrent class.
Consider the Markov chain in Example 11.6.a. Is Class 1 = {state 1, state 2} aperiodic?b. Is Class 2 = {state 3, state 4} aperiodic?c. Is Class 4 = {state 6, state 7, state 8} aperiodic?Example
Let {N(t), t ∈ [0,∞)} be a Poisson process with rate λ. Show the following: given that N(t) = n, the n arrival times have the same joint CDF as the order statistics of n independent Uniform(0,
For the Markov chain given in Figure 11.12, answer the following questions: How many classes are there? For each class, mention if it is recurrent or transient. -13 1 ² 2 3 Figure 11.12 - A state
Let {N(t), t ∈ [0,∞)} be a Poisson process with rate λ. Let T1, T2, ⋯ be the arrival times for this process. Find E[T₁+T₂ + +T₁0|N(4) = 10].
Consider the Markov chain in Figure 11.12. Let's define bi as the absorption probability in state 3 if we start from state i. Use the above procedure to obtain bi for i = 0, 1, 2, 3.
Consider a Markov chain with two possible states, S = {0, 1}. In particular, suppose that the transition matrix is given bywhere a and b are two real numbers in the interval [0, 1] such that 0 where
Consider the Markov chain shown in Figure 11.13. Let tk be the expected number of steps until the chain hits state 1 for the first time, given that X0 = k. Clearly, t1 = 0. Also, let r1 be the mean
Two teams A and B play a soccer match. The number of goals scored by Team A is modeled by a Poisson process N1(t) with rate λ1 = 0.02 goals per minute, and the number of goals scored by Team B is
In Problem 10, find the probability that Team B scores the first goal. That is, find the probability that at least one goal is scored in the game and the first goal is scored by Team B.Problem 10Two
Consider a Markov chain in Example 11.12: a Markov chain with two possible states, S = {0, 1}, and the transition matrixwhere a and b are two real numbers in the interval [0, 1] such that 0 0 and r1,
Let {N(t), t ∈ [0,∞)} be a Poisson process with rate λ. Let p : [0,∞) ↦ [0, 1] be a function. Here we divide N(t) to two processes N1(t) and N2(t) in the following way. For each arrival, a
Let α0, α1, ⋯ be a sequence of nonnegative numbers such thatConsider a Markov chain X0, X1, X2, ⋯ with the state space S = {0, 1, 2,⋯} such thatShow that X1, X2, ⋯ is a sequence of i.i.d
Consider the Markov chain with three states S = {1, 2, 3}, that has the state transition diagram is shown in Figure 11.31.Suppose P(X1 = 1) = 1/2 and P(X1 = 2) = 1/4.a. Find the state transition
Consider a Markov chain in Example 11.12: a Markov chain with two possible states, S = {0, 1}, and the transition matrixwhere a and b are two real numbers in the interval [0, 1] such that 0 Example
Consider the Markov chain shown in Figure 11.14.a. Is this chain irreducible?b. Is this chain aperiodic?c. Find the stationary distribution for this chain.d. Is the stationary distribution a limiting
Consider the Markov chain in Figure 11.32. There are two recurrent classes, R1 = {1, 2}, and R2 = {5, 6, 7}. Assuming X0 = 4, find the probability that the chain gets absorbed to R1.
Consider the Markov chain of Problem 16. Again assume X0 = 4. We would like to find the expected time (number of steps) until the chain gets absorbed in R1 or R2. More specifically, let T be the
Consider a continuous Markov chain with two states S = {0, 1}. Assume the holding time parameters are given by λ0 = λ1 = λ > 0. That is, the time that the chain spends in each state before
Consider the Markov chain shown in Figure 11.15. Assume that 0 < p < 1/2. Does this chain have a limiting distribution?
Consider the Markov chain shown in Figure 11.33. Assume X0 = 2, and let N be the first time that the chain returns to state 2, i.e.,Find E[N|X0 = 2]. N = min {n ≥ 1: X₂ = 2}.
Consider the continuous 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
Real-life systems often are composed of several components. For example, a system may consist of two components that are connected in parallel as shown in Figure 1.28. When the system's components
Consider the Markov chain shown in Figure 11.35.This is known as the simple random walk. Show thatUsing Stirling's formula, it can be shown thatis finite if and only if p ≠ 1/2. Thus, we conclude
Consider a continuous-time Markov chain X(t) that has the jump chain shown in Figure 11.23. Assume the holding time parameters are given by λ1 = 2, λ2 = 1, and λ3 = 3. Find the limiting
Consider the Markov chain shown in Figure 11.34.a. Is this chain irreducible?b. Is this chain aperiodic?c. Find the stationary distribution for this chain.d. Is the stationary distribution a limiting
You would like to go from point A to point B in Figure 1.28. There are 5 bridges on different branches of the river as shown in Figure 1.29.Bridge i is open with probability Pi, i = 1, 2, 3, 4, 5.
You choose a point (X,Y ) uniformly at random in the unit square S = {(x, y) ∈ R2 : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}.Let A be the event {(x, y) ∈ S : |x −y| ≤ 1/2} and B be the event {(x, y) ∈
One way to d esign a spam filter is to look at the words in an email. In particular, some words are more frequent in spam emails. Suppose that we have the following information:50% of emails are
You are in a game show, and the host gives you the choice of three doors. Behind one door is a car and behind the others are goats. You pick a door, say Door 1. The host who knows what is behind the
I toss a fair die twice, and obtain two numbers X and Y . Let A be the event that X = 2, B be the event that X +Y = 7, and C be the event that Y = 3.a. Are A and B independent?b. Are A and C
You and I play the following game: I toss a coin repeatedly. The coin is unfair and P(H) = p. The game ends the first time that two consecutive heads (HH) or two consecutive tails (TT) are observed.
A box contain s two coins: a regular coin and one fake two-headed coin (P(H)=1). I choose a coin at random and toss it n times. If the first n coin tosses result in heads, what is the probability
A family has n children, n ≥ 2. We ask the father: "Do you have at least one daughter?" He responds "Yes!" Given this extra information, what is the probability that all n children are girls? In
A family has n children, n ≥ 2. We ask from the father, "Do you have at least one daughter named Lilia?" He replies, "Yes!" What is the probability that all of their children are girls? In other
A family has n children. We pick one of them at random and find out that she is a girl. What is the probability that all their children are girls?
A coffee shop has 4 different types of coffee. You can order your coffee in a small, medium, or large cup. You can also choose whether you want to add cream, sugar, or milk (any combination is
Suppose that I want to purchase a tablet computer. I can choose either a large or a small screen; a 64GB, 128GB, or 256GB storage capacity, and a black or white cover. How many different options do I
I need to choose a password for a computer account. The rule is that the password must consist of two lowercase letters (a to z) followed by one capital letter (A to Z) followed by four digits (0,
Eight committee members are meeting in a room that has twelve chairs. In how many ways can they sit in the chairs?
There are 20 black cell phones and 30 white cell phones in a store. An employee takes 10 phones at random. Find the probability thata. There will be exactly 4 black cell phones among the chosen
Let A be a set with |A| = n
Five cards are dealt from a shuffled deck. What is the probability that the dealt hand containsa. Exactly one ace;b. At least one ace?
If k people are at a party, what is the probability that at least two of them have the same birthday? Suppose that there are n = 365 days in a year and all days are equally likely to be the birthday
Five cards are dealt from a shuffled deck. What is the probability that the dealt hand contains exactly two aces, given that we know it contains at least one ace?
Shuffle a deck of 52 cards. How many outcomes are possible? (In other words, how many different ways can you order 52 distinct cards? How many different permutations of 52 distinct cards exist?)
The 52 cards in a shuffled deck are dealt equally among four players, call them A, B, C, and D. If A and B have exactly 7 spades, what is the probability that C has exactly 4 spades?
I choose 3 cards from the standard deck of cards. What is the probability that these cards contain at least one ace?
There are 50 students in a class and the professor chooses 15 students at random. What is the probability that you or your friend Joe are among the chosen students?
Show the following identities for non-negative integers k and m and n, using combinatorial interpretation arguments. 1. We have 2-0 (2) = 2². k=0 2. For 0 ≤ k
How many distinct sequences can we make using 3 letter "A"s
In how many ways can you arrange the letters in the word "Massachusetts"?
You have a biased coin for which P(H) = p. You toss the coin 20 times. What is the probability thata. You observe 8 heads and 12 tails;b. You observe more than 8 heads and more than 8 tails?
A wireless sensor grid consists of 21 ×11 = 231 sensor nodes that are located at points (i, j) in the plane such that i ∈ {0, 1,⋯, 20} and j ∈ {0, 1, 2,⋯, 10} as shown in Figure 2.1. The
Suppose that I have a coin for which P(H) = p and P(T) = 1 −p. I toss the coin 5 times.a. What is the probability that the outcome is THHHH?b. What is the probability that the outcome is HTHHH?c.
In Problem 10, assume that all the appropriate paths are equally likely. What is the probability that the sensor located at point (10, 5) receives the message? That is, what is the probability that a
Ten people hav e a potluck. Five people will be selected to bring a main dish, three people will bring drinks, and two people will bring dessert. How many ways they can be divided into these three
In Problem 10, assume that if a sensor has a choice, it will send the message to the above sensor with probability pa and will send the message to the sensor to the right with probability pr = 1
I roll a die 18 times. What is the probability that each number appears exactly 3 times?
Ten passengers get on an airport shuttle at the airport. The shuttle has a route that includes 5 hotels, and each passenger gets off the shuttle at his/her hotel. The driver records how many
There are two coins in a bag. For Coin 1, P(H) = 1/2 and for Coin 2, P(H) = 1/3. Your friend chooses one of the coins at random and tosses it 5 times.a. What is the probability of observing at least
There are 15 people in a party, including Hannah and Sarah. We divide the 15 people into 3 groups, where each group has 5 people. What is the probability that Hannah and Sarah are in the same group?
You roll a die 5 times. What is the probability that at least one value is observed more than once?
I have 10 red and 10 blue cards. I shuffle the cards and then label the cards based on their orders: I write the number one on the first card, the number two on the second card, and so on. What is
I have two bags. Bag 1 contains 10 blue marbles, while Bag 2 contains 15 blue marbles. I pick one of the bags at random, and throw 6 red marbles in it. Then I shake the bag and choose 5 marbles
How many distinct solutions does the following equation have such that all xi ∈ N? x1 + x2 + x3 + x4 + x5 = 100 €
In a communication system, packets are transmitted from a sender to a receiver. Each packet is received with no error with probability p independently from other packets (with probability 1 −p the
How many distinct solutions does the following equation have? x1 + x₂ + x3 + x4 = 100, such that x₁ € {0, 1,2,,10}, x2, x3, x4 € {0, 1, 2, 3,...}.
For this problem suppose that the xi's must be non-negative integers, i.e., xi ∈ {0, 1, 2,⋯} for i = 1, 2, 3. How many distinct solutions does the following equation have such that at least one
I toss a coin five times. This is a random experiment and the sample space can be written as S = {TTTTT,TTTTH,...,HHHHн}. =
Find the range for each of the following random variables.1. I toss a coin 100 times. Let X be the number of heads I observe.2. I toss a coin until the first heads appears. Let Y be the total number
Let X be the number of the cars being repaired at a repair shop. We have the following information:At any time, there are at most 3 cars being repaired.The probability of having 2 cars at the shop is
I toss a fair coin twice, and let X be defined as the number of heads I observe. Find the range of X, RX, as well as its probability mass function PX.
I roll two dice and observe two numbers X and Y . If Z = X −Y , find the range and PMF of Z.
Let X and Y be two independent discrete random variables with the following PMFs:anda. Find P(X ≤ 2 and Y ≤ 2).b. Find P(X > 2 or Y > 2).c. Find P(X > 2|Y > 2).d. Find P(X
I have an unfair coin for which P(H) = p, where 0 < p < 1. I toss the coin repeatedly until I observe a heads for the first time. Let Y be the total number of coin tosses. Find the distribution of Y .

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