Question

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.

Answer #1

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.
Derive the joint probability distribution function for X and Y.
Make sure to explain your steps.

Let
X be a binomial random variable with parameters n = 500 and p =
0.12. Use normal approximation to the binomial distribution to
compute the probability P (50 < X ≤ 65).

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 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

The random variables X and Y are independent.
X has a Uniform distribution on [0, 5], while Y
has an Exponential distribution with parameter λ = 2. Define W
= X + Y.
A. What is the expected value of
W?
B. What is the standard deviation of
W?
C. Determine the pdf of
W. For full credit, you need to write out
the integral(s) with the correct limits of integration. Do not
bother to calculate the integrals.

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?

Independent random variables X and Y follow binomial
distributions with parameters(n1,θ) and (n2,θ). Let Z =X+Y. What
will be the distribution of Z?
Hint: Use moment generating function.

The random variable X has a Binomial distribution with
parameters n = 9 and p = 0.7
Find these probabilities: (see Excel worksheet)
Round your answers to the nearest hundredth
P(X < 5)
P(X = 5)
P(X > 5)

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).

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