Question

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 .

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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
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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?

Let X be an exponential random variable with parameter λ > 0.
Find the probabilities P( X > 2/ λ ) and P(| X − 1 /λ | < 2/
λ) .

Let X denote a random variable that follows a binomial
distribution with parameters n=5, p=0.3, and Y denote a random
variable that has a Poisson distribution with parameter λ = 6.
Additionally, assume that X and Y are independent random
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Make sure to explain your steps.

Let X~Poisson(4) random variable and Y an independent
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(a) Use Markov's inequality to find an upper bound for P(X+Y
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(b) Use Chebyshev's inequality to find an upper bound for P(X+Y
> 15)

Suppose X and Y are independent Poisson random variables with
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Let two random variables X and Y satisfy Y |X = x ∼ Poisson (λx)
for all possible values x from X, with λ being an unknown
parameter. If (x1, Y1), ...,(xn,
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x, construct the estimator for λ using the method of maximum
likelihood and determine its unbiasedness.

Let X denote a random variable that follows a binomial
distribution with parameters n=5, p=0.3, and Y denote a random
variable that has a Poisson distribution with parameter λ = 6.
Additionally, assume that X and Y are independent random
variables.
Using the joint pdf function of X and Y, set up the summation
/integration (whichever is relevant) that gives the expected value
for X, and COMPUTE its value.

Let X denote a random variable that follows a binomial
distribution with parameters n=5, p=0.3, and Y denote a random
variable that has a Poisson distribution with parameter λ = 6.
Additionally, assume that X and Y are independent random
variables.
Using the joint pdf function of X and Y, set up the summation
/integration (whichever is relevant) that gives the expected value
for X, and COMPUTE its value.

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