Question

- Poisson Distribution: p(x,
λ) = λ
^{x}exp(-λ) /x! , x = 0, 1, 2, …..

- Find the moment generating function M
_{x}(t)

- Find E(X) using the moment generating function

2. If X_{1} , X_{2} ,
X_{3} are independent and have means 4, 9, and
3, and variencesn3, 7, and 5. Given that Y =
2X_{1} - 3X_{2} +
4X_{3.} find the

- mean of Y
- variance of Y.

3. A safety engineer claims that 2 in 12 automobile accidents are due to driver fatigue. Using the formula for Binomial Distribution find the probability that at least 3 of 5 automobile accidents are due to driver fatigue.

4. The number of complaints that a dry-cleaning establishment receives per day is a random variable having Poisson distribution with λ = 1.8. Use Poisson distribution formula to find the probability that there will be at most two complaints on any given day.

5.

i. Define Geometric distribution.

- Show that it is a distribution

- Find the moment generating function for Geometric distribution

Answer #1

Given the exponential distribution f(x) = λe^(−λx), where λ >
0 is a parameter. Derive the moment generating function M(t).
Further, from this mgf, find expressions for E(X) and V ar(X).

The moment generating function for the random variable X is
MX(t) = (e^t/ (1−t )) if |t| < 1. Find the variance of X.

(i) If a discrete random variable X has a moment generating
function
MX(t) = (1/2+(e^-t+e^t)/4)^2, all t
Find the probability mass function of X. (ii) Let X and Y be two
independent continuous random variables with moment generating
functions
MX(t)=1/sqrt(1-t) and MY(t)=1/(1-t)^3/2, t<1
Calculate E(X+Y)^2

Suppose that the moment generating function of a random variable
X is of the form MX (t) = (0.4e^t + 0.6)8 . What is the moment
generating function, MZ(t), of the random variable Z = 2X + 1?
(Hint: think of 2X as the sum two independent random variables).
Find E[X]. Find E[Z ]. Compute E[X] another way - try to recognize
the origin of MX (t) (it is from a well-known distribution)

Use the moment generating function Mx(t) to find the mean u and
variance o^2. Do not find the infinite series.
Mx(t) = e^[5*((e^t)-1)]

The random variable X has moment generating function
ϕX(t)=exp((9t)^2)/2)+15t)
Provide answers to the following to two decimal places
(a) Evaluate the natural logarithm of the moment generating
function of 2X at the point t=0.62.
(b) Hence (or otherwise) find the expectation of 2X.
c) Evaluate the natural logarithm of the moment generating
function of 2X+7 at the point t=0.62.

A Poisson distribution has λ = 4.7. (a
)[2] Use the Excel function POISSON.DIST() and 5 decimals to
fill in the following table: 4 x 0 1 2 3 4 5 6 7 8 9 10 11 12+
P(x)
(b)[2] Use a column chart to visualize the probability
distribution above. How is it skewed?
(c)[1] Find P(x ≤ 3). [steps & result]
(d)[1] Find P(x ≥ 7). [steps & result]
(e)[2] Find P(5 < x ≤ 9). [steps & result]...

Given a random variable X~exp(5). Z=(X-2)3
1. Find the distribution function FZ(t).
2. Find fz(t).

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