Question

Suppose X1 and X2 are independent expon(λ) random variables. Let Y = min(X1, X2) and Z = max(X1, X2).

(a) Show that Y ∼ expon(2λ)

(b) Find E(Y ) and E(Z).

(c) Find the conditional density fZ|Y (z|y).

(d) FindP(Z>2Y).

Answer #1

Suppose that X1 and X2 are independent continuous random
variables with the same probability density function as: f(x) = ( x
2 0 < x < 2, 0 otherwise. Let a new random variable be Y =
min(X1, X2,).
a) Use distribution function method to find the probability
density function of Y, fY (y).
b) Compute P(Y > 1).
c) Compute E(Y )

Let X1, X2, X3 be independent random variables, uniformly
distributed on [0,1]. Let Y be the median of X1, X2, X3 (that is
the middle of the three values). Find the conditional CDF of X1,
given the event Y = 1/2. Under this conditional distribution, is X1
continuous? Discrete?

Suppose that X1 and X2 are independent continuous random
variables with the same probability density function as: f(x) = ( x
2 0 < x < 2, 0 otherwise. Let a new random variable be Y =
min(X1, X2,).
a) Use distribution function method to find the probability
density function of Y, fY (y).
b) Compute P(Y > 1).

Let X1, X2,... be a sequence of
independent random variables distributed exponentially with mean 1.
Suppose that N is a random variable, independent of the Xi-s, that
has a Poisson distribution with mean λ > 0. What is the expected
value of X1 + X2 +···+
XN2?
(A) N2
(B) λ + λ2
(C) λ2
(D) 1/λ2

Suppose that X1 and X2 are independent standard normal random
variables. Show that Z = X1 + X2 is a normal random variable with
mean 0 and variance 2.

Let Y be the liner combination of the independent random
variables X1 and X2 where Y = X1 -2X2
suppose X1 is normally distributed with mean 1 and standard
devation 2
also suppose the X2 is normally distributed with mean 0 also
standard devation 1
find P(Y>=1) ?

Let X1 and X2 be independent random variables such that X1 ∼ P
oisson(λ1) and X2 ∼ P oisson(λ2). Find the distribution of Y = X1 +
X2.s

X1 and X2 are iid exponential (2) random
variables and Z=max(X1 , X2). What is
E[Z]?
(Hint: Find CDF and then PDF of Z)
A. 3/2
B. 3
C. 1/2
D. 3/4

Consider n independent variables, {X1, X2, . . . , Xn} uniformly
distributed over the unit interval, (0, 1). Introduce two new
random variables, M = max (X1, X2, . . . , Xn) and N = min (X1, X2,
. . . , Xn).
(A) Find the joint distribution of a pair (M, N).
(B) Derive the CDF and density for M.
(C) Derive the CDF and density for N.
(D) Find moments of first and second order for...

You are given that X1 and X2 are two independent and identically
distributed random variables with a Poisson distribution with mean
2. Let Y = max{X1, X2}. Find P(Y = 1).

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