Question

Let X_{1} and X_{2} be independent Poisson
random variables with respective parameters λ_{1} and
λ_{2.} Find the conditional probability mass function
P(X_{1} = k | X_{1} + X_{2} = n).

Answer #1

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

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, 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

Let X1 and X2 be two independent geometric
random variables with the probability of success 0 < p < 1.
Find the joint probability mass function of (Y1,
Y2) with its support, where Y1 =
X1 + X2 and Y2 =
X2.

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, 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?

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)

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