Question

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

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

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).

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 }.

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)

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

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).

Suppose X and Y are independent Poisson random variables with
respective parameters λ = 1 and λ = 2. Find the conditional
distribution of X, given that X + Y = 5. What distribution is
this?

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

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