Let X and Y be integrable random variables on the probability space (Ω,F,P)andA be a sub-σ-field of F. Show that (i) if X ≤ Y a.s., then E(X|A) ≤ E(Y|A) a.s.;

Q: Show that the probability for two random polynomials in Z[r] of degree at most n and max-norm at…

A: If  аn  аssessment  is  essentiаl  оr  рerсeived  benefiсiаl  in  аny  mаnner,  аuthоrs  аre  likely…

Q: Let f : Z10 → Z10 be a random permutation. What is the probability that f (0) = 0 and f(1) = 1?…

A: Let f:ℤ10→ℤ10 be a random permutation.  Suppose A be the event that f0=0 B be the event that f1=1 To…

Q: (a) Let X and Y be two random variables possibly defined on two different probability spaces…

A:

Q: Let (2, A, P) be a probability space, let X be an integrable random variable. Let C be a…

A: @solution:::

Q: Let X and Y be an arbitrary integrable random variables on (N, F, P), and let Z(w) = aX(w) + bY (w),…

A: From the given information, X and Y are arbitrary integrable random variables. Let Zω=aXω+bYω, a,…

Q: Theorem 8.3.4 Let u -x(m) and V- x(n) and let U and V be inde- pendent. Then the random variable U/m…

A: Answer: From the given data

Q: If both X and Y are continuous random variables, then for any interval (a, b) CR and constant c E R,…

A: Joint probability is the probability of two events happening together.

Q: Theorem 4. Let X be a discrete random variable with probability function p(x), g1(X), g2(X), g3(X),…

A:

Q: If X is a random variable, prove that Cov(X,X) = σX².

A: Let X is a random variable.

Q: Let X be a discrete random variable with probability mass function e-1 P(X= k) = , k = 0,1,2,... k!…

A: From the given information, the probability mass function for X is,

Q: Let Y be a discrete random variable. The probability mass function (pmf) of Y, p(y), must satisfy…

A: HERE USE CONDITION OF VALID PROBABILITY MASS FUNCTION

Q: LetX bea discrete random variable withprobability functionp(x), g1(X),g2(X),g3(X),···,gk(X) be k…

A:

Q: 5- Let X; ,i = 1,2,3 independent random varibles such that X, Poisson(i),Vi = 1,2,3 then P (Σ 1 X…

A: Solution: Let Xi, i=1,2,3  are independent random variables.  Xi~Poisson(λi=i)     ∀ i= 1,2,3…

Q: Prove that cov(X, Y) = cov(Y, X) for both discreteand continuous random variables X and Y.

A:

Q: 3. Let fx.y(x, y) be the joint probability function for the random vector (X, Y). Which of the…

A: A joint probability is the probability of two events happening together.

Q: Let X and Y be independent random variables. Prove that f(X) and g(Y) are independent random…

A:

Q: Suppose that X, and X2 are aiscrete random variables with joint pdf of the form f(x1,x2) = c(x1 +…

A:

Q: Let X be a discrete random variable with probability mass function e-1 ,k=0,1,2,.. k! P(X= k) = Then…

A:

Q: Suppose that f(x, y) = (1/12 1/6 1/3 0 , (1,2), (3,2) , (2,2), (3,3), (1,3) , (2,4) , otherwise is a…

A:

Q: Let random variable X be uniform in the interval (0, 1). Define random variable Y = aX + b where a…

A: Given that X ~ U0,1 and Y=aX+b such that a≠0

Q: Let the random variables (discrete) X1, X2 have the joint pmf p(x1, x2) = 1/3, for (x1, 12) =…

A: Covariance is given by, Cov(X1,X2)= E(X1)*E(X2)-E(X1*X2) E(X1)= (summation X1*p(X1,X2)) =…

Q: If X is a continuous random variables, then Covs.) (a) o (b) , (c) P (d)) None of them as- If X and…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: 2.| Let X1,..., X„ be n random variables. Prove that max (X;) = X; -Emin (X;, X;) + i<j i=1 + E…

A: It is given that X1,..., Xn are n random variables. The objective is to prove that…

Q: X be a standard normal random variable, X ∼ N(0, 1). Find a real-valued scalar k such that: E[(aX^2…

A: Given: X ~ N (0, 1) EaX2+bk=1 a + b &lt; 1 where k is a real number and a and b are positive real…

Q: . If there are n random variables such that each expectation exist then prove that E(X1+…….+Xn  ) =…

A: We need to prove given theorem for discrete random variables

Q: Consider a probability space (N, F, P) and a random variable X e L-0, F, P). solution to only b…

A: Given: Let's determine EEXGH=EXH

Q: Let fx.x(z, y) be the joint probability function for the random vector (X,Y) (discrete or…

A: Given f(x,y) be joint probability function for the random vector X and Y. We have to conclude that…

Q: Let X, Y, and Z be jointly distributed random variables. Prove that Cov(X + Y, Z) = Cov(X, Z) +…

A: The covariance between two variables is given as follows (provided the expectation of random…

Q: Let {Xn} and {Yn} be sequences of random variables such that Xn diverges to o in probability and Yn…

A:

Q: Let X and Y be independent random variables. 1. Are X and -Y independent? (Hint: Use Theorem: Let…

A:

Q: 6.3. Prove or disprove: for any random variable Y defined on 2 = {H,T}N and any probability measure…

A: For any Random variable Y defined on Ω = H, Tn and any probability measure P defined on Ω. The…

Q: Cousider a datnset for binnry clnssificntion problem with class The fentures are given by F1, F7 and…

A: Initially we have to find the frequency and probability table for each class: Class F1:   C0 C1…

Q: Consider a probability space (N, F, P) and a random variable XE L(0, F, P). (a) Show that the…

A: Given Probability Space (Ω,F,P) Random Variable X∈L1[Ω,F,P] Find Show that the collection of random…

Q: Let X be continuous with pdf f(x)=e* if 0 < x < ∞, and zero otherwise: . Use Chebychev's inequality…

A: Introduction: For an exponentially distributed random variable X, with mean λ, the probability…

Q: 2. Let X be a discrete random variable whose range is the nonnegative integers. Show that E[X]=…

A: # discrete random variable x with non negative integers  Then to prove that E(x)=sum(1-F(x))  F(x)…

Q: The number of customers arriving at a bank between 11:00 a.m. and 3:00 p.m. follows a Poisson law…

A: Given : The number of customers arriving at a bank follows a Poisson distribution λper hour = 20

Q: Theorem 4. Let X be a discrete random variable with probability function p(x), 91(X), 92(X), g3(X),…

A:

Q: a. The Central Limit Theorem is valid regardless whether the random variables X_i that are added up…

A: Here in the central limit theorem X's are must be independent and sum of all observations.…

Q: 6) Show that, if X and Y are independent random variables, such that E|X| < co ,and B is an…

A:

Q: Which of the stochastic matrice in Exercise is regular?

A: We know that : A matrix is regular if its determinant is non zero. A stochastic matrix is regular…

Q: Prove that the maximal entropy of a discrete random variable is logan (n being the number of…

A:

Q: (b) Let X and Y be independent random variables. Prove that f(X) and g(Y) are independent random…

A:

Q: 7. Explain the concept of a random vector ana now it induces a probabity on the sets of the Borel…

A: We need to, Explain the concept of a random vector How random vector can induce probability on the…

Q: 2. Let X1, X2, X3 ., X, be iid b(1, p) random variables. Let S, = E=1 X¢ then prove that Sn-E(Sn)…

A: From the given information, X1, X2,.....,Xn be iid b1,p random variables. The probability mass…

Q: 4. Let X1 and X2 be discrete random variables with the joint multinomial distirbution x2 2-x1-x2 2…

A:

Q: Let XXX be uniformly distributed on [1,3][1,3][1,3]. What is the expectation of X2X^2X2, i.e.,…

A: We have to find the expectation of X2, i.e., E[X2]. Where X be uniformly distributed on [1,3].

Q: 3. Let (N, F, P) be a probability space and (Xn)n>1 an iid sequence of random variables such that 1…

A: Xn P(Xn=1) P(Xn=2) P(Xn=3) P(Xn=4) X 1/10 2/10 3/10 4/10 (Xn)n≥1 be a…

Q: If random variables X1 and X2 are independent then .A o=(°x'x)aq »(x,x,)="," cov (X ,X_)=0 cov 2 .B…

A: Given,the random varibles X1 , X2 are independent .

Q: Let X be a discrete random variable with PMF: (0.3 for k = 0 0.3 for k = 1 Px (k) = {0.4 for k = 2…

A: For  X=0⇒Y=0X=1⇒Y=0X=2⇒Y=0X=3⇒Y=6   There fore Y assumes only two values 0  for X=0.1.2 &amp; 6 for…

Q: Let {Xn} and {Yn} be sequences of random variables such that Xn diverges to ∞ in probability and Yn…

A: