43. Log-normal distribution. Let X be N (0, 1), and let Y = eX; Y is said to have the log-normal distribution. Show that the density function of Y is 1 f(x)=√exp{-}(log x)²}, x > 0. For la ≤ 1, define fa (x) = {1+a sin(2л log x)} f(x). Show that fa is a density function with finite moments of all (positive) orders, none of which depends on the value of a. The family {fa: a ≤ 1} contains density functions which are not specified by their moments.

Q: If f(x) = tan x, which of the equations below describes the average value of f(x) over the interval…

A: We have to find the average value of the given function. This problem can be solved by using formula…

Q: The first derivative of a continuous function y=f(x) is given. Find y' and then sketch the general…

A: The first derivative of a continuous function y=f(x) is given.

Q: 2. Find the average value of the function f(x) = sin(x) on [0, 27]. Could you have guessed the…

A: Given function is

Q: Use logarithmic differentiation to find dy/dx. y = (x + 1)(x-3) x>3 Step 1 The given function y is…

A:

Q: Suppose X1 and X2 are iid observations from the pdf f(x]a) = ax°-le-",x > 0, a > 0. Show that (log…

A: see the attachment.

Q: Integrate the following functions f(x) = 4/2/201 22²2 + + [2,³] X 3 x

A:

Q: Find the equation of normal to the function f(x)=x3−3x+1 at its local maximum point. I want a HUMAn…

A: The objective of this question is to find the equation of the normal line to the function f(x) = x^3…

Q: 38. The function g is defined by g(x) = x In(x). What is the average value of g on the interval [1,…

A:

Q: Let X ~ exp(rate = 3). (a) Find the mgf of X. (b) Use the mgf to find E[X], Var(X), E[X4] and E[X").

A:

Q: Inx 2. Let f be the function given by f(x) = for all x > 0. The derivative offis given by 1- ln x…

A: The given function is; f(x)=lnx/x

Q: Let f(x) = e*g(x), where g(0) = 1 and g'(0) = 4. Find f'(0). %3D %3D

A: fx=exgx Differentiating with respect to x, we get f'x=dexdx·gx+dgxdx·ex…

Q: Suppose f(x) = cos x. Without using the AROC formula or a calculator, find the average rate of…

A:

Q: Given that f(z) = z'h(z) h(- 1) = 3 h'(-1) = 6 %3D Calculate f'(- 1).

A:

Q: Let f be a function with f(5) = -2 and whose derivative is If f '(x) = 5x - 9. %3D %3D А. Write a…

A:

Q: Estimate the true percent relative error for the first derivative of the function, y = e*cos(x?), at…

A: