(1) X ∼ N(0, 1), define T = X^(−2) . Find the p.d.f. of T
(2) X ∼ N(µ, σ^2 ). Define Y = X^2/σ^2. Find the p.d.f. of Y and the m.g.f. of Y
Let X have the p.d.f. f(x) = 3(1−x)2 for 0 < x < 1. Find p.d.f. of Y = (1−X)3.
Let, f(x)= 1/15e-x/15, 0≤ x < ∞ be the p.d.f. of X i. find the c.d.f., F(x), for f(x). ii. find the values of µ and ?2. iii. what is the moment generating function? iv. what is the probability that 20<x<40? v. what percentile is µ? vi. what is the value of the 25th percentile?
1. Suppose that the p.d.f. of a random variable X is as follows: for 0<x<2, for 0 〈 x 〈 2. r for 0<< f(x) = 0 otherwise. Let Y - X (2 - X). First determine the c.d.f. of Y, then find its p.d.f. (Hint: when computing c.d.f., plotting the function Y- X(2 - X) which may help. )
4. Let X have p.d.f. fx(1),-1 < 2. Find the p.d.f. of Y-X2
Problem 1 (11 pts] The independent r.v.'s X and Y have p.d.f. f(t) = et, t>0. Compute the probability: P(X+Y > 2). Hint: Use independence of X and Y in order to find their joint p.d.f., fx,y, and then use the diagram below to compute the probability: P(X+Y < 2). y 2 r+y = 2 y . ! 2 0 2-y Note: If X and Y represent the lifetimes of 2 identical equipment of expected lifetime 1 time unit, then...
A random sample of size n = 2 is taken from the p.d.f
f(x) = {1 for 0 ≤ x ≤ 1 and 0 otherwise.
Find P(X-bar ≥ 0.9)
3. A random sample of size n = 2 is taken from the p.d.f 1 for 0 < x < 1 f(30 0 otherwise. Find P(X > 0.9)
1. Let the joint p.d.f of X and Y be 2xe if 0 < x < 1 and y > x2 fxy(z, y) 0, otherwise. (a) Find the marginal p.d.f.'s of X and Y, respectively (b) Compute P(Y < 2X2)
1. Let the joint p.d.f of X and Y be 2xe if 0
Find a consistent estimator of µ 2 , where E(Y ) = µ is the
population mean and Y¯ n is the sample mean. 2 If E(Y 2 ) = µ 0 2
then prove that 1 n Pn i=1 Y 2 i is an consistent estimator of µ 0
2 3 We define σ 2 = µ 0 2 − µ 2 . Show that S 2 n = 1 n Pn i=1 Y 2
i − Y¯ 2...
qkx Q.3 (20 pts) The joint p.d.f. of 2 random variables x and y is given by f xy(x,y) = 05x,0 sy.(2x + y) = 2 to otherwise Where k is a constant. 1) (5 pts) Find the value of k. 2) (5 pts) Are x and y independent? Explain. 3) (10 pts) Define z = 2x-y. Find the p.d.f. of z.
2. Let Y = ex2, where X ~ Ņ(0, 1). (a) Find the p.d.f. of Y. J0 oPlY2 > ijdt. (b) C ompuite
2. Let Y = ex2, where X ~ Ņ(0, 1). (a) Find the p.d.f. of Y. J0 oPlY2 > ijdt. (b) C ompuite