Questions and Answers of Introduction To Probability Statistics

For the random variable Y in Example 3.4,Example 3.4I have an unfair coin for which P(H) = p, where 0 1. Check that ΣERY PY (y) = 1. Ry 2. If p, find P(2 ≤ Y
I toss a coin twice and define X to be the number of heads I observe. Then, I toss the coin two more times and define Y to be the number of heads that I observe this time. Find P(X 1) P((X < (Y > 1)).
N guests arrive at a party. Each person is wearing a hat. We collect all the hats and then randomly redistribute the hats, giving each person one of the N hats randomly. Let XN be the number of
Let X ∼ Binomial(n, p) and Y ∼ Binomial(m, p) be two independent random variables. Define a new random variable as Z = X + Y . Find the PMF of Z.
Suppose you take a pass-fail test repeatedly. Let Sk be the event that you are successful in your kth try, and Fk be the event that you fail the test in your kth try. On your first try, you have a 50
For each of the following random variables, find P(X > 5), P(2 < X ≤ 6) and P(X > 5|X < 8).
The number of emails that I get in a weekday can be modeled by a Poisson distribution with an average of 0.2 emails per minute.1. What is the probability that I get no emails in an interval of length
In this problem, we would like to show that the geometric random variable is memoryless. Let X ∼ Geometric(p). Show thatWe can interpret this in the following way: Remember that a geometric random
Let X be a discrete random variable with range RX = {1, 2, 3, . . . }. Suppose the PMF of X is given bya. Find and plot the CDF of X, FX(x).b. Find P(2 c. Find P(X > 4). Px (k) = for
An urn consists of 20 red balls and 30 green balls. We choose 10 balls at random from the urn. The sampling is done without replacement (repetition not allowed).a. What is the probability that there
Let X ∼ Bernoulli(p). Find EX.
The number of emails that I get in a weekday (Monday through Friday) can be modeled by a Poisson distribution with an average of 1/6 emails per minute. The number of emails that I receive on weekends
Let X ∼ Geometric(p). Find EX
Let X be a discrete random variable with the following CDF:Find the range and PMF of X. Fx(x) = 0 1 T 2 3 4 1 for x < 0 for 0 < x < 1 for 1 < x < 2 for 2 < x < 3 for x ≥ 3
Let X ∼ Poisson(λ). Find EX.
Let X be a discrete random variable with the following PMFa. Find EX.b. Find Var(X) and SD(X).c. If Y = 2/X, find EY . Px (k) = 0.5 0.3 0.2 0 for k = 1 for k = 2 for k= 3 otherwise
Let X ∼ Binomial(n, p). Find EX.
Let X be a discrete random variable with the following PMFThe random variable Y = g(X) is defined asFind the PMF of Y . Px(k)= = 21 0 for k € {-10, -9,,-1,0, 1,, 9, 10} otherwise
Let X ∼ Geometric(1/3), and let Y = |X −5|. Find the range and PMF of Y .
Let X ∼ Pascal(m, p). Find EX. (Try to write X = X1 + X2 +⋯+Xm, such that you already know EXi.)
Let X be a discrete random variable with rangesuch that ㅠ 3T Rx = {0, 1, 2, ³, }, 4 4
Let X be a discrete random variable with PX(k) = 1/5 for k = −1, 0, 1, 2, 3. Let Y = 2|X|. Find the range and PMF of Y.
Let X ∼ Pascal(m, p). Find V ar(X).
Prove E[aX +b] = aEX +b (linearity of expectation).
Suppose tha t Y = −2X +3. If we know EY = 1 and EY2 = 9, find EX and V ar(X).
I roll a fair die and let X be the resulting number. Find EX, Var(X), and σX.
There are 1000 households in a town. Specifically, there are 100 households with one member, 200 households with 2 members, 300 households with 3 members, 200 households with 4 members, 100
Suppose that there are N different types of coupons. Each time you get a coupon, it is equally likely to be any of the N possible types. Let X be the number of coupons you will need to get before
If X ∼ Binomial(n, p) find Var(X).
Let X be a random variable with mean EX = μ. Define the function f(α) asFind the value of α that minimizes f. f(a) = E[(X-a)²].
Here is a famous problem called the St. P etersburg Paradox. Wikipedia states the problem as follows: "A casino offers a game of chance for a single player in which a fair coin is tossed at each
The median of a random variable X is defined as any number m that satisfies both of the following conditions:Note that the median of X is not necessarily unique. Find the median of X ifa. The PMF of
You are offered to play the following game. You roll a fair die once and observe the result which is shown by the random variable X. At this point, you can stop the game and win X dollars. You can
I choose a real number uniformly at random in the interval [a, b], and call it X. By uniformly at random, we mean all intervals in [a, b] that have the same length must have the same probability.
Let X be a continuous random variable with the following PDFwhere c is a positive constant.a. Find c.b. Find the CDF of X, FX(x).c. Find P(1 fx(x) = { ce-x 0 x ≥ 0 otherwise
Choose a real number uniformly at random in the interval [2, 6] and call it X.a. Find the CDF of X, FX(x).b. Find EX.
Let X be a continuous random variable with the following PDFwhere c is a positive constant.a. Find c.b. Find the CDF of X, FX(x).c. Find P(2 d. Find EX. fx(x) = {: 0 ce 4x x ≥ 0 otherwise
Let X be a continuous random variable with PDFa. Find E(Xn), for n = 1, 2, 3,⋯.b. Find the variance of X. fx(x) = √ √ ₂² + ²/ x2 { 10 0 ≤ x ≤ 1 otherwise
Let X ∼ Uniform(a, b). Find EX.
Let X be a continuous random variable with PDFFind the expected value of X. fx(x) {} 0 2x 0≤x≤1 otherwise
Let X be a continuous random variable with PDFFind E(Xn), where n ∈ N. √ - {₁ {ő 0 fx(x) = + 1/1/202 x + 0≤x≤1 otherwise
Let X be a uniform(0, 1) random variable, and let Y = e−X.a. Find the CDF of Y.b. Find the PDF of Y.c. Find EY.
Let X be a continuous random variable with PDFand let Y = X2.a. Find the CDF of Y.b. Find the PDF of Y.c. Find EY. fx(x): = 32 0 0 < x≤2 otherwise
Let X be a continuous random variable with PDFFind the mean and variance of X. fx(x) = 0 x ≥ 1 otherwise
Let X ∼ Exponential(λ), and Y = aX, where a is a positive real number. Show that Y ~ Exponential (à).
Let X be a Uniform(0, 1) random variable, and let Y = eX.a. Find the CDF of Y.b. Find the PDF of Y.c. Find EY.
Let X ∼ Exponential(λ). Show that
Let X ∼ Uniform(−1, 1) and Y = X2. Find the CDF and PDF of Y.
Let X be a continuous random variable with PDFand let Y = 1/X. Find fY (y). 1={1,2² 4x³ 0 fx(x) 0 < x < 1 otherwise
Let X ∼ N(3, 9).a. Find P(X > 0).b. Find P(−3 < X < 8).c. Find P(X > 5|X > 3).
Let X be a continuous random variable with PDFand let Y = X2. Find fY(y). fx(x) = 1 2πT for all x ER
Let X ∼ N(3, 9) and Y = 5 −X.a. Find P(X > 2).b. Find P(−1 < Y < 3).c. Find P(X > 4|Y < 2).
Let X be a continuous random variable with PDFand let Y =√|X̄|. Find fY(y). 1 fx(x) = e /27 for all x E R.
Let X ∼ N(−5, 4).a. Find P(X < 0).b. Find P(−7 < X < −3).c. Find P(X > −3|X > −5).
Answer the following questions:1. Find Γ(7/2).2. Find the value of the following integral: T = 1 * I 2°e52dư.
Let X ∼ Exponential(2) and Y = 2 +3X.a. Find P(X > 2).b. Find EY and V ar(Y ).c. Find P(X > 2|Y < 11).
The median of a continuous random variable X can be defined as the unique real number m that satisfiesFind the median of the following random variablesa. X ∼ Uniform(a, b).b. Y ∼
Let X be a random variable with the following CDFa. Plot FX(x) and explain why X is a mixed random variable.b. Find P(X ≤ 1/3).c. Find P(X ≥ 1/4).d. Write CDF of X in the form of FX(x) = C(x)
Using the properties of the gamma function, show that the gamma PDF integrates to 1, i.e., show that for α,λ > 0, we have ∞ Xªxª-¹e-λx е r(a) So dx = 1.
Let X be a continuous random variable with the following PDF:Let alsoFind the CDF of Y. fx(x) 2x 0 0 ≤ x ≤ 1 otherwise
Let X be a random variable with the following CDFa. Find the generalized PDF of X, fX(x).b. Find EX using fX(x).c. Find V ar(X) using fX(x). Fx(x) = 0 8 x + 1 2 for x < 0 for 0 < x 1
Let Y be the mixed random variable defined in Example 4.14.Example 4.14:Let X be a continuous random variable with the following PDF:Let alsoFind the CDF of Y. a. Find P(≤Y ≤). b. Find
Consider two random variables X and Y with joint PMF given in Table 5.1.Figure 5.1 shows PXY (x, y).a. Find P(X = 0,Y ≤ 1).b. Find the marginal PMFs of X and Y .c. Find P(Y = 1|X = 0).d. Are X and
A surveillance system is in charge of detecting intruders to a facility. There are two hypotheses to choose from:H0: No intruder is present.H1: There is an intruder.The system sends an alarm message
Let X ∼ Bernoulli(p) and Y ∼ Bernoulli(q) be independent, where 0 < p, q < 1. Find the joint PMF and joint CDF for X and Y .
Let X and Y be the same as in Example 5.4.a. Find E[X|Y = 1|.b. Find E[X|−1c. Find E[|X||−1Example 5.4:Consider the set of points in the grid shown in Figure 5.4. These are the points in set G
Let X and Y be two independent Geometric(p) random variables. Find E X²+Y2- XY
I roll a fair die. Let X be the observed number. Find the conditional PMF of X given that we know the observed number was less than 5.
Consider two random variables X and Y with joint PMF given in Table 5.2. Let Z = E[X|Y].a. Find the Marginal PMFs of X and Y .b. Find the conditional PMF of X given Y = 0 and Y = 1, i.e., find PX|Y
Let X, Y, and Z = E[X|Y] be as in Example 5.11. Let also V =Var(X|Y).a. Find the PMF of V .b. Find EV.c. Check that Var(X) = E(V )+Var(Z).Example 5.11:Consider two random variables X and Y with joint
Let N be the number of customers that visit a certain store in a given day. Suppose that we know E[N] and Var(N). Let Xi be the amount that the ith customer spends on average. We assume Xi's are
In Problem 29, suppose that X and Y are independent Uniform(0, 1) random variables. Find the joint PDF of R and Θ. Are R and Θ independent?Problem 29Let X and Y be two independent standard normal
Let X and Y be two independent standard normal random variables. Consider the point (X,Y ) in the x −y plane. Let (R,Θ) be the corresponding polar coordinates as shown in Figure 5.11. The inverse
Let X and Y be two independent Uniform(0, 1) random variables. Find FXY (x, y).
Let X and Y be as in Example 5.24 in Section 5.2.3, i.e., suppose that we choose a point (X,Y ) uniformly at random in the unit disc Are X and Y uncorrelated?Example 5.24 in Section 5.2.3Consider
Let X and Y be two random variables with joint PDF fXY (x, y). Let Z = X +Y . Find fZ(z).
If Y ∼ Uniform(0, 1), find E[Yk] using MY(s).
Let Bn be the event that a graph randomly generated according to G(n, p) model has at least one isolated node. Show that P(Bn) > n(1-p)"-1. (2) (1 (1-p) 2-3
If X ∼ Binomial(n, p) find the MGF of X.
Consider the following random experiment: A fair coin is tossed once. Here, the sample space has only two elements S = {H,T}. We define a sequence of random variables X1, X2, X3, ⋯ on this sample
Let X1, X2, X3, ⋯ be independent random variables, where Xn ∼ Bernoulli (1 n) for n = 2, 3,⋯. The goal here is to check whether Xn a.s.→ 0.1. Check that2. Show that the sequence X1, X2, . . .
Let X1, X2, X3, . . ., Xn be a random sample. Show that the sample meanis an unbiased estimator of θ = EXi. Ô=X = X₁ + X₂+...+Xn n
Let X1, X2, X3, . . ., Xn be a random sample from the following distributionwhere θ ∈ [−2, 2] is an unknown parameter. We define the estimator ^Θn asto estimate θ.a. Is ^Θn an unbiased
Let X be a continuous random variable with the following PDFSuppose that we knowFind the posterior density of X given Y = 2, fX|Y (x|2). fx(x) = 6x(1-x) 0 if 0 ≤ x ≤ 1 otherwise
For the following random samples, find the maximum likelihood estimate of θ:1. Xi ∼ Binomial(3, θ), and we have observed (x1,x2,x3,x4) = (1, 3, 2, 2).2. Xi ∼ Exponential(θ) and we have
Consider a communication channel as shown in Figure 9.2. We can model the communication over this channel as follows. At time n, a random variable Xn is generated and is transmitted over the channel.
Suppose that you would like to estimate the portion of voters in your town that plan to vote for Party A in an upcoming election. To do so, you take a random sample of size n from the likely voters
Let X be a continuous random variable with the following PDFAlso, suppose thatFind the MAP estimate of X given Y = 5. fx(x) = 3x² 0 if 0 ≤ x ≤ 1 otherwise
Let X and Y be two jointly continuous random variables with joint PDFFind the MAP and the ML estimates of X given Y = y. fxy(x, y) = x+2y² 0 0≤x, y ≤ 1 otherwise.
Let X be a continuous random variable with the following PDF:Also, suppose thatFind the MAP estimate of X given Y = 3. fx(x) = 2x { 0 if 0 ≤ x ≤ 1 otherwise
Let X ∼ Uniform(0, 1). Suppose that we know Y | X = x ∼ Geometric(x).Find the posterior density of X given Y = 2, fX|Y (x|2).
Let X be a continuous random variable with the following PDFWe also know thatFind the MMSE estimate of X, given Y = y is observed. fx(x) = 2x² + 0 3 if 0 ≤ x ≤ 1 otherwise
Let X ∼ N(0, 1) andwhere W ∼ N(0, 1) is independent of X.a. Find the MMSE estimator of X given Y , ( X̂M).b. Find the MSE of this estimator, using MSE = E[(X −X̂M)2].c. Check that E[X2] = E[
Suppose that the signal X ∼ N(0,σ2X) is transmitted over a communication channel. Assume that the received signal is given by Y = X +W, where W ∼ N(0,σ2W) is independent of X.1. Find the ML
Let X be a continuous random variable with the following PDFWe also know thatFind the MMSE estimate of X, given Y = y is observed. fx(x) = 2x 0 if 0 ≤ x ≤ 1 otherwise
Suppose X ∼ Uniform(0, 1), and given X = x, Y ∼ Exponential (λ = 1/2x).a. Find the linear MMSE estimate of X given Y .b. Find the MSE of this estimator.c. Check that E[X̃Y ] = 0.
Suppose that the signal X ∼ N(0,σ2X) is transmitted over a communication channel.Assume that the received signal is given by Y = X +W, where W ∼ N(0,σ2W) is independent of X.a. Find the MMSE
Let X ∼ N(0, 1) and Y = X +W, where W ∼ N(0, 1) is independent of X.a. Find the MMSE estimator of X given Y , ( X̂M).b. Find the MSE of this estimator, using MSE = E[(X −X̂M)2].c. Check that
Let X be an unobserved random variable with EX = 0, Var(X) = 5. Assume that we have observed Y1 and Y2 given byY1 = 2X +W1,Y2 = X +W2,where EW1 = EW2 = 0, Var(W1) = 2, and Var(W2) = 5. Assume that
Consider again Problem 8, in which X is an unobserved random variable with EX = 0, Var(X) = 5. Assume that we have observed Y1 and Y2 given byY1 = 2X +W1,Y2 = X +W2,where EW1 = EW2 = 0, Var(W1) = 2,

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