Question
In: Statistics and Probability
(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 the probability mass function for X. Then determine the mean and variance.
Solutions
orchestra answered 1 year agoRelated Solutions
(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...
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...
Suppose that a continuous, positive random variable T has a hazard function defined by h(t) =...
Suppose that a continuous, positive random variable T
has a hazard function defined by
h(t) = hj when
aj-1 ≤ t <
aj, j = 1, 2, …,
k,
where a0 = 0 and ak= ∞,
a0 < a1 < … <
ak and
hj> 0, j = 1, 2, …, k.
(a) Express S(t), the survivor function of
T, in terms of the hj’s.
(b) Express f(t), the probability density
function of T, in terms of the
hj’s.
(c)...
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).
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).
If the moment-generating function of X is M(t) = exp(3 t + 12.5 t2) = e3...
If the moment-generating function of X is M(t) = exp(3 t + 12.5
t2) = e3 t + 12.5 t2.
a. Find the mean and the standard deviation of
X.
Mean =
standard deviation =
b. Find P(4 < X < 16). Round your answer
to 3 decimal places.
c. Find P(4 < X2 < 16). Round
your answer to 3 decimal places.
If the moment generating function is Mx(t) = 1/(1-10t) for t near 0 whats the pdf,...
If the moment generating function is Mx(t) = 1/(1-10t) for t
near 0
whats the pdf, mean, and variance with steps please want to
understand
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...
Let T > 0 be a continuous random variable with cumulative hazard function H(·). Show that...
Let T > 0 be a continuous random variable with cumulative
hazard function H(·).
Show that H(T)∼Exp(1),where Exp(λ) in general denotes the
exponential distribution with rate parameter λ.
Hints:
(a) You can use the following fact without proof: For any
continuous random variable T with cumulative distribution function
F(·), F(T) ∼ Unif(0,1). Hence S(T) ∼ Unif(0,1).
(b) Use the relationship between S(t) and H(t) to derive that
Pr(H(T) > x)= e−x.
2) Suppose that you need to generate a random variable Y with a density function f...
2) Suppose that you need to generate a random variable Y with a
density function f (y) corresponding to a beta distribution with
range [0,1], and with a non-integer shape parameter for the beta
distribution. For this case there is no closed-form cdf or inverse
cdf. Suppose your choices for generating Y are either: a) an
acceptance-rejection strategy with a constant majorizing function
g(u) = V over [0, 1], i.e., generate u1 and u2 IID from a U[0,1]
generator and...
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