a.
Prove that the moment-generating
a.
Explanation of Solution
From the given information, X follows Poisson distribution with parameter
The moment generating function of X is
Then,
Hence proved
b.
Prove that
b.
Explanation of Solution
From the given information, the expansion is
From the part a
Then,
As
Hence proved.
c.
Prove that the distribution function of Y converges to a standard
c.
Explanation of Solution
From the theorem 7.5, if Y and
From the part b, as
This is the moment generating function of the standard normal distribution.
By using theorem 7.5,
Hence proved.
Want to see more full solutions like this?
Chapter 7 Solutions
Mathematical Statistics with Applications
- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.arrow_forwardLet m(t) be the moment generating function of a random variable X. Show that the random variable W = 10X is m(10t). What is the moment generating function of Z = X-5 in terms of m(t)?arrow_forwardTheorem 6.4 states that the moment-generating function of the gamma distribution is given by Mx(t) = (1-βt)^(-α).arrow_forward
- Let X be a continuous random variable with pdf -2 < x< 4 fx(x) = otherwise Give the moment generating function (mgf) of X, mx(t).arrow_forward(21) Let X b(12,–) find E(5+6x) and distribution function.arrow_forwardIf X be a continuous random variable with -bx be f(x)= x >0 if otherwise then the moment generating function of X is given by b -t (b/1 b +tarrow_forward
- 3) Let X be a continuous random variable with PDF 1 x20 fx(x) ={0 otherwise Find E[X], Var[X] and Mx(t) = . E[e*].arrow_forwardLet X and Y be jointly continuous random variables with joint PDF given by: fx₁y (x,y) = = X (1+3y² ) / (0₁2) (X) I (01) (y) 4 a. Find the conditional PDF of X given y b. Find P ( ₁ < x < 1/2 | Y = £) 3 c. Find the marginal PDF of X.arrow_forwardSuppose that Xi ∼ Gamma(αi , β) independently for i = 1, . . . , N. The mgf(moment generating function) of Xiis MXi(t) = (1 − (t/β) )−αi . (a)Use the mgf of Xi to derive the mgf of ∑i=1 Xi . Determine the distribution of ∑i=1 Xi based on its mgf.arrow_forward
- Q. 3 Let continuous random variables X and Y be independent identically distributed random variables with the following respective pdfs: fx(x) = 2e-2x;x 2 0 and fy (y) = 2e-2y;y > 0 Define a new random variable Z = X + Y. Derive moment generating function of Z, i.e. Mz(t).arrow_forwardLet X be a random variable with the moment generating function M(t) = 1/(1 – t)?, t < 1. Find E(X³) and Var(X).arrow_forwardIf X has an exponential distribution, show thatP[(X Ú t + T)|(x ÚT)] = P(X Ú t)arrow_forward
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,