Question

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?

Answer #1

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Let X and Y be independent random variables following Poisson
distributions, each with parameter λ = 1. Show that the
distribution of Z = X + Y is Poisson with parameter λ = 2. using
convolution formula

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 }.

Suppose X, T are independent exponential random variables with
parameters λX and λT . Find the conditional density of X given X
< T .

Let X follow Poisson distribution with λ = a and Y follow
Poisson distribution with λ = b. X and Y are independent. Define a
new random variable as Z=X+Y. Find P(Z=k).

Let X be a Poisson random variable with parameter λ and Y an
independent Bernoulli random variable with parameter p. Find the
probability mass function of X + Y .

Let X and Y be independent random variables each having the
uniform distribution on [0, 1].
(1)Find the conditional densities of X and Y given that X > Y
.
(2)Find E(X|X>Y) and E(Y|X>Y) .

a. Suppose X and Y are independent Poisson
random variables, each with expected value 2. Define Z=X+Y. Find
P(Z?3).
b. Consider a Poisson random variable X with
parameter ?=5.3, and its probability mass function, pX(x). Where
does pX(x) have its peak value?

Problem 0.1
Suppose X and Y are two independent exponential random variables
with respective densities given by(λ,θ>0) f(x) =λe^(−xλ) for
x>0 and g(y) =θe^(−yθ) for y>0.
(a) Show that Pr(X<Y) =∫f(x){1−G(x)}dx {x=0, infinity] where
G(x) is the cdf of Y, evaluated at x [that is,G(x) =P(Y≤x)].
(b) Using the result from part (a), show that P(X<Y)
=λ/(θ+λ).
(c) You install two light bulbs at the same time, a 60 watt bulb
and a 100 watt bulb. The lifetime of the...

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).

Topic: Linear Combination Of Random Variables
Suppose X and Y are independent random variables with X ∼ N(1,
9) and Y ∼ N(2, 16). Find the probability that 2Y ≥ 1; find the
probability that X − Y ≥ 0.

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