PROOF Let X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE If X = Y, then Cov(X,Y) = Var(Y)
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PROOF
Let X and Y be continuous random variables with joint distribution function, F (x,y).
Let g (X,Y) and h (X,Y) be
PROVE
If X = Y, then Cov(X,Y) = Var(Y)
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- PROOF Let X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE E[g(X,Y) + h(X,Y)] = E[g(X,Y)] + E[h(X,Y)]PROOF Let X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE Var(aX + bY) = a2 Var(X) + b2 Var(Y) + 2ab Cov (X,Y)PROOF Let X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE Var(aX) = a2 Var (X)
- PROOF Let X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE If X and Y are independent, then E[XY] = E[X] E[Y]Proof Let X and Y be independent integrable random variables on a probability space and f be a nonnegative convex function. Show that E[f(X +Y)] ≥ E[f(X +EY)].Proof Let X be a random variable with EX2 < ∞ and let Y = |X|. Suppose that X has a Lebesgue density symmetric about 0. Show that X and Y are uncorrelated, but they are not independent.
- Proof. Let X be a random variable and let g(x) be a non-negative function. Then for r>0, P [g(X) ≥ r] ≤ Eg(X)/rProof Let X and Y be integrable random variables on the probability space (Ω,F,P)andA⊂Fbe a σ-field. Show that E[YE(X|A)] = E[XE(Y|A)], assuming that both integrals exist.Proof Show thata randomvariable Xis independent ofit selfifan donlyif Xis constanta.s.CanXandf(X)beindependent,wherefisa Borelfunction?
- Probability Model If the probability density of a random variable is given by F(x)= x, 2-x,0 for 0<x<1, for 1≤x<2 , elsewhere Find the probabilities that a random variable having this probability density will take on a value Between 0.2 and 0.8;(b) 0.6 and 1.2Probability density function of the random variable X (pdf) fX(x) = ( c(1 − x2) −a ≤ x ≤ a ( 0 d.y get given in the form. Event |X − b| < a/2 defined as. variables a, b to probability rules randomly give the appropriate values. After determining the values of a, b; Under the condition A of the random variable X Find the probability density function.Exercise 2: Let X be a random density variable f(x) = { e−2θx if x ≥ 0 0 else Find θ so that f is a probability density. Determine the distribution function F of the variable X. Calculate the expectation and variance of X. Let Y be the random variable defined by Y = θX. What is the law of Y?
