Problem 3. Let X be a discrete random variable, gx) - a+ bX+ cX, and let...
Verify the linearity of expectation: if X is a discrete random variable (with a finite range), and its expectation is defined as where f is the probability mass function of X. prove that E = [X+Y]=E[X]+E[Y],andE[cX]=cE[X] for any real number c. E[x] => + f(x) T
3 (2) (x)(Ax Bx), (Ex)(Cx Bx), (x)(CXAX) (Ex) (Gx Hx) (3) (Ex)(Gx Fx), (ax)Fx, (Ex) Gx 3x) Fx (4) (x)(Fx Gx) (2) (x)(Ax Bx), (Ex)(Cx Bx), (x)(CXAX) (Ex) (Gx Hx) (3) (Ex)(Gx Fx), (ax)Fx, (Ex) Gx 3x) Fx (4) (x)(Fx Gx)
Let X be a discrete random variable with probability function f(x). Prove that E[a + b g(X) + c h(X)] = a + bE[g(X)] + cE[h(X))], where g and h are functions, and a, b and c are constants.
Problem 4 Let X be a discrete random variable with probability mass function fx(x), and let t be a function. Define Y = t(X): that is, Y is the randon variable obtained by applying the function t to the value of X Transforming a random variable in this way is frequently done in statistics. In what follows, let R(X) denote the possible values of X and let R(Y) denote the possible values of To compute E[Y], we could irst find...
Problem 5. Prove the following result for any number a and discrete random variable X. 티(X-a 21 = Var(X) + (E(X)-a)2 You must start your proof by using the definition of the expected value of a function of a discrete random variable, i.e. where g(x)- (x-a)
Problem 3. Let X be a discrete random variable, with probability distribution Determine x1 and X2 such that E(X-0 and ơ2(X-7.
Problem 3. Let X be a discrete random variable, with probability distribution P(X X1) = 0.95, P(X X2) = 0.05. Determine x, and X2 such that E(X-0 and σ2(X) = 7.
Let X be a discrete random variable with a probability mass function (pmf) of the following quadratic form: p(x) = Cx(5 – x), for x = 1,2,3,4 and C > 0. (a) Find the value of the constant C. (b) Find P(X ≤ 2).
3. Let X be a continuous random variable with probability density function ax2 + bx f(0) = -{ { for 0 < x <1 otherwise 0 where a and b are constants. If E(X) = 0.75, find a, b, and Var(X). 4. Show that an exponential random variable is memoryless. That is, if X is exponential with parameter > 0, then P(X > s+t | X > s) = P(X > t) for s,t> 0 Hint: see example 5.1 in...
Problem 4 Let X be the following discrete random variable: Let Y = X2. Show that cov(X·Y) = 0, but X and Y are not independent random variable.