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 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 }.

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

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

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, Y be random variables. Also let X|Y = y ~ Poisson(y) and
Y ~ gamma(a,b) is the prior distribution for Y. a and b are also
known.
1. Find the posterior distribution of Y|X=x where X=(X1, X2, ...
, Xn) and x is an observed sample of size n from the distribution
of X.
2. Suppose the number of people who visit a nursing home on a
day is Poisson random variable and the parameter of the Poisson...

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?

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