Let X and Y be two independent random variables. X is a binomial
(25,0.4) and Y is a uniform (0,6). Let W=2X-Y and Z= 2X+Y.
a) Find the expected value of X, the expected value of Y, the
variance of X and the variance of Y.
b) Find the expected value of W.
c) Find the variance of W.
d) Find the covariance of Z and W.
d) Find the covariance of Z and W.
(a) Let X be a binomial random variable with parameters (n, p).
Let Y be a binomial random variable with parameters (m, p).
What is the pdf of the random variable Z=X+Y?
(b) Let X and Y be indpenednet random variables. Let Z=X+Y.
What is the moment generating function for Z in terms of those
for X and Y?
Confirm your answer to the previous problem (a) via moment
generating functions.
Let x be a random variable that possesses a binomial
distribution with p=0.5 and n=9. Using the binomial formula or
tables, calculate the following probabilities. Also calculate the
mean and standard deviation of the distribution. Round solutions to
four decimal places, if necessary.
P(x≥3)=
P(x≤8)=
P(x=5)=
μ=
σ=
Suppose X and Y are two independent random variables with
X~N(1,4) and Y~N(4,6).
a. P(X < -1.5).
b. P(0.5Y+1 > 5).
c. P(-2 < X + Y < 5).
d. P(X – Y ≥ 3).
Let X be a binomial random variable with n =
11 and p = 0.3. Find the following values. (Round your
answers to three decimal places.)
(a)
P(X = 5)
(b)
P(X ≥ 5)
(c)
P(X > 5)
(d)
P(X ≤ 5)
(e)
μ = np
μ =
(f) σ =
npq
σ =
Let X and Y be two independent random variables. Assume that X
is Negative-
Binomial(2, θ) and Y is Negative-Binomial(3, θ) distributed. Let Z
be another random
variable, Z = X + Y .
(a) Find the following probabilities: P(Z = 0), P(Z = 1) and P(Z =
2);
(b) Can you guess what is the distribution of Z?
9.8 Let X and Y be independent random variables with probability
distributions given by
P(X = 0) = P(X = 1) = 1/2 and P(Y = 0) = P(Y = 2) = 1/2 .
a. Compute the distribution of Z = X + Y .
b. Let Y˜ and Z˜ be independent random variables, where Y˜ has
the same distribution as Y , and Z˜ the same distribution as Z.
Compute the distribution of X˜ = Z˜ − Y
Let X represent a binomial random variable with
n = 110 and p = 0.19. Find the following
probabilities. (Do not round intermediate calculations.
Round your final answers to 4 decimal places.)
a.
P(X ≤ 20)
b.
P(X = 10)
c.
P(X > 30)
d.
P(X ≥ 25)