Question

7.5) If X_{1} and X_{2} are independent random
variables having exponential densities with the parameters
θ_{1} and θ_{2}, use the distribution function
technique to find the probability density of
Y=X_{1}+X_{2} when

a) θ_{1} ≠ θ_{2}

b) θ_{1} = θ_{2}

7.7) With reference to the two random variables of Exercise 7.5,
show that if θ_{1} = θ_{2} = 1, the random variable
Z_{1}=X_{1}/(X_{1} + X_{2}) has the
uniform density with α=0 and β=1_{.}

**(I ONLY NEED TO ANSWER 7.7)**

Answer #1

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)

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

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 and X2 be two independent random variables having gamma
distribution with parameters α1 = 3, β1 = 3 and α2 = 5, β2 = 1,
respectively. We are interested in finding the distribution of Y =
2X1 + 6X2. A standard approach is to apply a two-step procedure as
that in question 2. However, as we discussed in the class, if the
MGF technique is applicable, then it would be preferred due to its
simplicity.
(a) Find the...

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 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}.

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, . . . Xn be iid
exponential random variables with unknown mean β. Find the method
of moments estimator of β

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 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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