Problem 8. Suppose that XGeom(p) and Y ~ Geom(r) are independent. Find the probability 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
If X and Y have a joint probability density function specified by 2-(+2y) find P(X <Y).
o Additional Problem 3: Suppose both X and Y are independent and distributed according to Geo(0.2). Compute P(min (X, Y Hint: If X ~ Geo(p), then FX (k) = 1-(1-pt. < 4).
Suppose that NoP(X5)6 and P(X 2):2 find P( 3< X < 4).
. 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)
. Let X and Y be the proportion of two random variables with joint probability density function f(x, y)o, elsewhere. (a) Find P(X < 3|Y= 2). (b) Are X and Y independent? Why? (c) Find E(Y/X)
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.
Assume that X and Y are independent and follow normal distributions with Hx (a) evaluate P(X +Y > 24) (2pt) (b) that P(z < X-Y < 10) = 0.2 (3pt) find r such
Let X N(1,3) and Y~ N(2,4), where X and Y are independent 1. P(X <4)-? P(Y < 1) =? 4、 5, P(Y < 6) =? 7, P(X + Y < 4) =?
is independent of X, and e Problem 3 Suppose X N(0, 1 -2) -1 <p< 1. (1) Explain that the conditional distribution [Y|X = x] ~N(px, 1 - p2) (2) Calculate the joint density f(x, y) (3) Calculate E(Y) and Var(Y) (4) Calculate Cov(X, Y) N(0, 1), and Y = pX + €, where