t X and Y be independent random variables with variance ơ1 and ơ3. respectively. Consider the...
Please solve 6.66. Thank you in advanced. 6-66 Letxand y be independent random variables with variances o2 and o2,respectively. Consider the sum z=ax + (1-a)y 0-as l Find a that minimizes the variance of z.
Let X, Y be two independent exponential random variables with means 1 and 3, respectively. Find P(X> Y)
Suppose that the standard normal random variables X and Y are independent. Find P(0 < X<Y). 8 O 1 4T 0 1 8л Ala
3. Suppose that X and Y are independent exponentially distributed random variables with parameter λ, and further suppose that U is a uniformly distributed random variable between 0 and 1 that is independent from X and Y. Calculate Pr(X<U< Y) and estimate numerically (based on a visual plot, for example) the value of λ that maximizes this probability.
2. Let X and Y be independent, exponentially distributed random variables where X has mean 1/λ and Y has mean 1/μ. (a) What is the joint p.d.f of X and Y? (b) Set up a double integral for determining Pt <X <Y) (c) Evaluate the above integral. (d) Which of the following equations true, and which are false? {Z > t} = {X > t, Y > t} (e) Compute P[Z> t) wheret 0. (f) Compute the p.d.f. of Z.
The answer will be: Consider two independent random variables X and Y. Let fx(x) 1-2 if 0 〈 x 〈 2 2-2y for 0-y 〈 l and 0 otherwise. Find the probability density function of X + Y. and 0 otherwise. Let Jy(y) 42.4 If 0-a-l then Íx+y(a) = 2a--a2+ ). If 1- a < 2 then 3 213 then jx+Y(a then fx+y(a)
4) The random variables X and Y have the joint PDF fx,y(x, y) = 0 < x < 6,0 < y < 6 Find E [X2Y2].
3. Consider two random variables X and Y, whose joint density function is given as follows. Let T be the triangle with vertices (0,0), (2,0), and (0,1). Then if (x, y for some constant K (a) (2 pts.) Find the constant K (b) (4 pts.) Find P(X +Y< 1) and P(X > Y). (c) (4 pts.) Find the marginal densities fx and fy. Conclude that X and Y are not independent
Suppose the joint pdf of random variables X and Y is f(x,y) = c/x, 0 < y < x < 1. a) Find constant c that makes f (x, y) a valid joint pdf. b) Find the marginal pdf of X and the marginal pdf of Y. Remember to provide the supports c) Are X and Y independent? Justify
. Let X and Y be the proportion of two random variables with joint probability density function f(r, y) e-*, 0, if, 0 < y < x < oo, elsewhere. a) Find P(Xc3.y-2). b) Are X and Y independent? Why? c) Find E(Y/X)