Question: If we let KX(t) = lnMX (t), the coefficient of tr/r! in the Maclaurins series of KX(t) is called the rth cumulant, and it

If we let KX(t) = lnMX – µ(t), the coefficient of tr/r! in the Maclaurin’s series of KX(t) is called the rth cumulant, and it is denoted by kr. Equating coefficients of like powers, show that
(a) k2 = µ2;
(b) k3 = µ3;
(c) k4 = µ4 – 3µ22.

Step by Step Solution

★★★★★

3.39 Rating (168 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

M x e t M x t K x t t ln M x t M x t 1 ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

583-S-J-P-D (882).docx

120 KBs Word File

Students Have Also Explored These Related Statistics Questions!

With reference to Exercise 6.39, show that for normal distributions k2= 2 and all other cumulants are zero. In exercise If we let KX(t) = lnMX (t), the coefficient of tr/r! in the Maclaurins series...

If we let RX(t) = lnMX(t), show that R'X(0) = and R''X(0) = 2. Also, use these results to find the mean and the variance of a random variable X having the moment- generating function MX(t) = e4(e4 ...

If we let a = µ in the first part of Theorem 4.10 on page 128, where µ is the mean of X, we get (a) Show that the rth derivative of MX µ(t) with respect to t at t = 0 gives the rth...

2.4 D Worksheet Chain Rule with Tables & Graphs Name: X -3 2 7 8 g(x) 3 2 h(x) 7 -2 g'(x) -1 5 2 8 h'(x) -1 7 -3 -5 Selected values of the differentiable functions g and h along with their...

Find the truth of the expression. Let P be the real numbers. Let K ( x , y ) = x

14.4 Kernel stability. Suppose an approximation of the kernel matrix K, denoted K0, is used to train the hypothesis h0 (and let h denote the non-approximate hypothesis). At test time, no...

Let f(x) RR be an infinitely differentiable function. Let k(x) be the smallest integer that satisfies f(j) (x) = 0, 1 j k 1, (k) (x) 0 (that is the smallest index of the derivative that is nonzero)....

Please help with this question 6. Let V be a finite dimensional vector space over a field F, and let R, T E C(V). Suppose that SE C(V) is invertible, and that T = SRS-1. Let Kx (T) denote the...

17. Given y = sin2 x Find the equation for the tangent and normal lines to the graph where x = E. 18. Given f(x) = sin2 x and g(x) = x2 5. Let H(x) = f(g(x)) and K06) = g(f(x)). a. Find H'(x) and...

1. (01.07 MC) sin x Let k(x) be a piecewise defined function so that k(x) = X if * #0 The function h(x) is given by the graph below. Which of the following equals lim ? (1 point) 0 if x = 0 x-0 K(x)...

What is the purpose of recruitment advertising? Why is it under the domain of corporate advertising and public relations?

Use the graph of f '(x) to sketch the graph of f(x). Sketch f(x) on the coordinate plane below f(x) f'(x) 44y -4-3-2-1 3- -2+ -3 T 2 3 4

Current monthly payment is $ 2 3 5 6 . The original term of the mortgage was 3 0 years and the mortgage is exactly 4 years and eight months. The mortgage interest rate is 6 . 3 7 5 % APR. how much do...

The historical growth record shows that economic growth as we know it today started well before 1750.

Yearly automobile inspections are required for residents of the state of Pennsylvania. Suppose that 18% of all inspected cars in Pennsylvania have problems that need to be corrected. Unfortunately,...

Consider again the landowners decision problem described in Problem 39. Suppose now that, at a cost of $90,000, the landowner can request that a soundings test be performed on the site where natural...

The chief executive officer of a firm in a highly competitive industry believes that one of her key employees is providing confidential information to the competition. She is 90% certain that this...

HARTSELL MANUFACTURING HARTSELL MANUFACTURING puts much emphasis on cash flow when it plans for capital investments. The company chose its discount rate of 8% based on the rate of return it must pay...

Assessment Description Using the INVOICE table structure shown in Table 1 , do the following: Table 1 : Attribute Name Sample Value Sample Value Sample Value Sample Value Sample Value INV _ NUM 1 0 0...

Kokomochi is considering the launch of an advertising campaign for its latest dessert product, the Mini Mochi Munch. Kokomochi plans to spend $5.6 million on TV, radio, and print advertising this...