Question

If X1 and X2 are independent exponential random variables with respective parameters λ1 and λ2, find the distribution of Z = min{X1, X2}.

Answer #1

Let X1,X2,..., Xn be independent random variables that are
exponentially distributed with respective parameters λ1,λ2,...,
λn.
Identify the distribution of the minimum V =
min{X1,X2,...,Xn}.

Let X1 and X2 be independent Poisson
random variables with respective parameters λ1 and
λ2. Find the conditional probability mass function
P(X1 = k | X1 + X2 = n).

If X1 and X2 are independent exponential random variables with
respective parameters 1 and 2, find the distribution of Z = min{X1,
X2}.

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

7.5) If X1 and X2 are independent random
variables having exponential densities with the parameters
θ1 and θ2, use the distribution function
technique to find the probability density of
Y=X1+X2 when
a) θ1 ≠ θ2
b) θ1 = θ2
7.7) With reference to the two random variables of Exercise 7.5,
show that if θ1 = θ2 = 1, the random variable
Z1=X1/(X1 + X2) has the
uniform density with α=0 and β=1.
(I ONLY NEED TO ANSWER 7.7)

7.5) If X1 and X2 are independent random
variables having exponential densities with the parameters
θ1 and θ2, use the distribution function
technique to find the probability density of
Y=X1+X2 when
a) θ1 ≠ θ2
b) θ1 = θ2
7.7) With reference to the two random variables of Exercise 7.5,
show that if θ1 = θ2 = 1, the random variable
Z1=X1/(X1 + X2) has the
uniform density with α=0 and β=1.
(I ONLY NEED TO ANSWER 7.7)

Let X and Y be independent exponential random variables with
respective parameters 2 and 3.
a). Find the cdf and density of Z = X/Y .
b). Compute P(X < Y ).
c). Find the cdf and density of W = min{X,Y }.

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 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).

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

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 2 minutes ago

asked 11 minutes ago

asked 46 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago