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STAT 150 Homework 23. Random Variable X takes integer values and has the Moment Generating Function:...
STAT 150 Homework
23. Random Variable X takes integer values and has the Moment Generating Function: Mx(t)= 4/(2-e^t) - 6/(3-e^t).
1) What is the variance of X.
2) Find the probability P(X ≤ 2).
Solutions
orchestra answered 2 years agoRelated Solutions
STAT 150 Homework 23. Random Variable X takes integer values and has the Moment Generating Function:...
STAT 150 Homework
23. Random Variable X takes integer values and has the Moment
Generating Function: Mx(t)=
4/(2-e^t) - 6/(3-e^t).
Find the probability P(X ≤ 2).
Two random variable X; Y has the following joint moment generating function MX;Y (s; t) =...
Two random variable X; Y has the following joint moment
generating function
MX;Y (s; t) = 0:5 + 0:1et +
0:1es + 0:1es+t + 0:15e2s +
0:05e2s+t
Find the probability of Y = 1 given X < 2
Hint:
> Find the PGF
> Deduce the joint probability mass function PMF
fX,Y(x,y)
>Plot the PMF on a 3-D plane x, y and z =
fX,Y(x,y) or alternatively use Cartesian (x-y) plane
with fX,Y(x,y) plots
>Use conditional Probability rues to obtain...
(a) Suppose that Y is a random variable with moment generating function H(t). Suppose further that...
(a) Suppose that Y is a random variable with moment generating
function H(t). Suppose further that X is a random variable with
moment generating function M(t) given by M(t) = 1/3 (2e ^(3t) +
1)H(t). Given that the mean of Y is 10 and the variance of Y is 12,
then determine the mean and variance of X (Use H(0) = 1).
(b) Suppose that the Moment generating function for X is M(t) =
e^t/( 3 − 2e^t) . Find...
(a) Suppose that Y is a random variable with moment generating function H(t). Suppose further that...
(a) Suppose that Y is a random variable with moment generating
function H(t). Suppose further that X is a random variable with
moment generating function M(t) given by M(t) = 1 3 (2e 3t +
1)H(t). Given that the mean of Y is 10 and the variance of Y is 12,
then determine the mean and variance of X (Use H(0) = 1).
(b) Suppose that the Moment generating function for X is M(t) =
e t 3 − 2e...
(a) Let X be a continuous random variable which only takes on positive values on the...
(a) Let X be a continuous random variable which only takes on
positive values on the interval [1, 4]. If P(X) = (√ x + √ 1 x )C 2
for all x in this interval, compute the value of C.
(b) Let X be a random variable with normal distribution. Let z
represent the z-score for X, and let a be a positive number. Prove
that P(z < |a|) = P(z < a) + P(z > −a) − 1.
Exercise: Variance of the uniform Suppose that the random variable X takes values in the set...
Exercise: Variance of the uniform
Suppose that the random variable X takes values in the set
{0,2,4,6,…,2n} (the even integers between 0 and 2n, inclusive),
with each value having the same probability. What is the variance
of X? Hint: Consider the random variable Y=X/2 and recall
that the variance of a uniform random variable on the set {0,1,…,n}
is equal to n(n+2)/12.
Var(X)=
Write a function convert_date that takes an integer as a parameter and returns three integer values...
Write a function convert_date that takes an integer as a
parameter and returns three integer values representing the input
converted into days, month and year (see the function
docstring).
Write a program named
t03.py that tests the function by
asking the user to enter a number and displaying the output day,
month and year. Save the function in a PyDev library module named
functions.py
A sample run for t03.py:
Enter a date in the format MMDDYYYY:
05272017
The output will...
Consider the two dependent discrete random variables X and Y . The variable X takes values...
Consider the two dependent discrete random variables X and Y .
The
variable X takes values in {−1, 1} while Y takes values in {1, 2,
3}. We observe that
P(Y =1|X=−1)=1/6
P(Y =2|X=−1)=1/2
P(Y =1|X=1)=1/2
P(Y =2|X=1)=1/4
P(X = 1) = 2/ 5
(a) Find the marginal probability mass function (pmf) of Y .
(b) Sketch the cumulative distribution function (cdf) of Y .
(c) Compute the expected value E(Y ) of Y .
(d) Compute the conditional expectation...
1. Suppose X has a binomial distribution with parameters n and p. Then its moment-generating function...
1. Suppose X has a binomial distribution with parameters n and
p. Then its moment-generating function is M(t) = (1 − p + pe^t ) n
.
(a) Use the m.g.f. to show that E(X) = np and Var(X) = np(1 −
p).
(b) Prove that the formula for the m.g.f. given above is
correct. Hint: the binomial theorem says that Xn x=0 n x a^x
b^(n−x) = (a + b)^n .
2. Suppose X has a Poisson distribution with...
Use moment generating functions to decide whether or not the given random variables X and Y...
Use moment generating functions to decide whether or not the
given random variables X and Y are equal in distribution.
a). The random variables Z1, Z2, Z3 are independent normal
N(0,1), X = Z1 + Z2 + Z3 and Y = √3Z1
b). The random variables Z1, Z2, Z3 are independent Poisson with
the same parameterλ, X = 3Z1 and Y=Z1 + Z2 + Z3
c). The random variables Z1, Z2 are independent normal N(0,1), X
= Z1 + 2Z2...
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