Question

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.

Answer #1

Let X, Y, and Z be independent and identically distributed
discrete random variables, with each having a probability
distribution that puts a mass of 1/4 on the number 0, a mass of 1/4
at 1, and a mass of 1/2 at 2.
a. Compute the moment generating function for S= X+Y+Z
b. Use the MGF from part a to compute the second moment of S,
E(S^2)
c. Compute the second moment of S in a completely different way,
by expanding...

Show that if two binomial random variables X ∼ Bin(a,p) and Y ∼
Bin(b,p) are independent, then X + Y ∼ Bin(a + b, p), using the
technique of moment generating function.

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

a) let X follow the probability density function f(x):=e^(-x) if
x>0, if Y is an independent random variable following an
identical distribution f(x):=e^(-x) if x>0, calculate the moment
generating function of 2X+3Y
b) If X follows a bernoulli(0.5), and Y follows a
Binomial(3,0.5), and if X and Y are independent, calculate the
probability P(X+Y=3) and P(X=0|X+Y=3)

Let ? and ? be two independent random variables with moment
generating functions ?x(?) = ?t^2+2t and
?Y(?)=?3t^2+t . Determine the
moment generating function of ? = ? + 2?. If possible, state the
distribution name (and include parameter values) of the
distribution of ?.

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 }.

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 two independent random vectors x and z have Gaussian
distributions: p(x) = N(x|µx,Σx), and p(z) = N(z|µz,Σz). Now
consider y = x + z. Use the results for Gaussian linear system to
ﬁnd the distribution p(y) for y. Hint. Consider p(x) and p(y|x).
Please prove for it rather than directly giving the result.

1) Let the random variables ? be the sum of independent Poisson
distributed random variables, i.e., ? = ∑ ? (top) ?=1(bottom) ?? ,
where ?? is Poisson distributed with mean ?? .
(a) Find the moment generating function of ?? . (b) Derive the
moment generating function of ?. (c) Hence, find the probability
mass function of ?.
2)The moment generating function of the random variable X is
given by ??(?) = exp{7(?^(?)) − 7} and that of ?...

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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