5.21 Given a probability space (S,F,P) and a random variable X. Let A be a sub-o-field of F, and consider  = E (X | A). (i) Show that E ((X – Â)Y) = 0 if Y is an A-measurable random vari- able. (ii) Let Y run over all A-measurable random variables. Show that the squared error E ((X – Y)²) is minimal when Y = X.

Q: Determine the value of c such that f(x) = (c / x2) for x > 1 represents the p.d.f. of a random…

A: Given that  f(x)=c/x2   , x>1

Q: Let Z be a continuous uniform random variable with E[Z] = 0. 10. Fill in the blanks Find the value…

A:

Q: Determine the value of k so that the function f(x)= k(x² + 4) may be used for x = 0,1,2,3, there is…

A: From the provided information, f(x) = k (x2 + 4) for x = 0, 1, 2, 3 there is a valid probability…

Q: 1 A probability density function of a random variable is given by f(x)=; =- 65 expected value, the…

A: A probability density function of a random variable is given by, f(x)=165 5+ 5x  ; 25≤x≤36…

Q: Let X, Y are random variables (r.v.) and a, b, c, d are values. Complete the formulas using the…

A: Step 1: Step 2: Step 3: Step 4:

Q: Let x be a random variable Gaussian with zero mean and variance 1. Find: a)The conditional pdf and…

A: To find the mean and Variance of positive half normal distribution

Q: Find fQ|X (q|x) for x ∈ {0, 1} and all q.

A: It is a fact of statistics. It is widely used.

Q: Exercise 3. Let X be a random variable with mean µ and variance o². For a E R, consider the…

A: Farmula used  V(X)=E(X2)-[E(X)]2

Q: (a) f(x) = c(x² + 6), for x = 0, 1, 2, 3 5 3 - (:) ( 2² X (b) f(x) = c 3-x for x = 0, 1, 2

A: It is given that a probability distribution of the discrete random variable X.

Q: Q3. The moment generating function for a random variable X is e3t M(t)=1-t2¹ What are the mean and…

A: The moment generating functionMx(t)=e3t/(1-t2)    -1<t<1To find mean and variance of X Concept…

Q: Let X be a random variable with p.d.f f(x) = c+ 1.2x2,0<x< 1. Find a constant c if E(X) = ! 0.6 0.4…

A:

Q: Let X be the continuous random variable which is the age of a first-year university student selected…

A:

Q: Let X and Y be dependent random variables. Which statements are always true? Select one or more:…

A: # Let X and Y be dependent random variables. Which statements are always true?

Q: Let X be a regular random variable, then V( E(X)) = OA. V(X). O B. some positive number. OC. EX).…

A: Random variables are of 2 types, discrete and continuous. A discrete random variable can only take…

Q: II. Let X be a discrete random variable with possible values xi, i=1,2,..., and p(xi)=P(X=xi) is…

A: The given statements can be completed as follows, Let X be a discrete random variable with possible…

Q: Let X denote the number of VCRs sold during a random week by a certain store. The probability…

A:

Q: Let X be a uniform random variable over the interval 1 to 4 and Y is exponential with a mean of 2.…

A: Given information: X~Uni(a=1,b=4) Standard deviation of X is: σX=b-a12       =4-112       =0.8660…

Q: et X be a discrete random variable with P(X = -2) = .1, P(X = 1) = .15, P(X = 2) = .4, P(X = 4) =…

A:

Q: Determine the value of k such that this will be a probability mass function for x = 17, 19, 21, 23:…

A: To determine the value of k such that f(x) is a probability mass function (PMF), we need to ensure…

Q: (Math 161A/163 review) Suppose that X₁, X2, X3 are independent Exponential (A) random variables. A>…

A:

Q: Find the Mean of f(x), where f(x) = 6X+6. The probability distribution of a random variable X ; if X…

A: Mean is an average value around which the whole data set is concentrated.…

Q: f(x) 1/6 2/6 a) Calculate the cumulative distribution of f. b) Find P(X<3). c) Find E(X) and ox. 2/6…

A: Given that Probability mass function of X is x 1 2 3 4 f(x) 1/6 2/6 2/6 1/6

Q: X is a discrete random variable with p.m.f given by P(X=x)= x/c where c is a constant. The support…

A:

Q: x is a uniform random variable in the interval (x1,x2). Find the expected value of x

A:

Q: ). Let M = ln(5*X) be a derived random variable. What is E[M]?

A:

Q: Solve the differential equations. ii. ду? 4x²y = e²y -

A: The given differential equation is                                     ∂2u∂y2-4x2y=e2y

Q: x < -1 Fx(x) = x/4+1/2 -1 < x < 1 %3D 1 Sketch the CDF and then find the following: a) P[X < 1] and…

A:

Q: Let X be a random variable with pdf given by fX(x) = 1/[π(1 + x2)] for all real number x. Prove that…

A: Solution

Q: The positive random variable X is said to be a log-normal random variable with parameters μ and σ2…

A:

Q: The probability distribution of the discrete random variable X is: f(X) = k23-x , for x 4,5,6, 7 )…

A: Hello! As you have posted more than 3 sub parts, we are answering the first 3 sub-parts.  In case…

Q: Let Z be a discrete random variable withE(Z)=0. Does it necessarily follow that E(Z3)=0? If yes,…

A: Solution

Q: Let X be a discrete random variable with P(X = -1) = .3, P(X = 1) = .45, P(X = 2) = .15 and P(X = 6)…

A: The following information has been given: We have given that X is a discrete random variable.  P(X =…