HOMEWORK VII

Due:  Monday, October 27

Improper Integrals

I   The Gamma Function

Many important functions in applied mathematics and statistics are defined in terms of improper integrals.  Perhaps the most famous of these is the gamma function, defined by:

This function is defined for x > 0.

(A)    Plot the graph of y = G(x) over the interval [0.5, 6].   (Let GammaFunction denote the improper integral above as an expression that depends upon t.)

(B)     By calculating G(1), G(2), G(3), G(4), and G(5), guess a formula for G(n) when n is a positive integer.

(C)     Calculate G(1/2), G(3/2) and G(5/2).

(D)      Show that G(x+1) = xG(x) for all x > 0.

(E)      Maple has a built-in function, GAMMA(x), that represents the gamma function given above.  This makes it easier for us to invoke the gamma function.  Stirling's formula states that

(Recall that two functions f and g are said to be asymptotic (and written f ~ g) if the limit of f(x)/g(x), as x tends toward infinity, equals 1.)  Using the formula for G(n) that you obtained in part (B), verify Stirling's formula by graphing the quotient of the two functions.

II  Normal Distribution

(A)    A function p(x) is said to be a probability density function (or pdf) if:

Show that each of the following functions is a probability density function.  Plot the graph of each pdf as well.

(1)    p1(x) =  (1/(3sqrt(2p)) exp(-(x-11)2/18)  defined for all x.
(2)    p2(x) = (1/sqrt(2p)) exp(-(x-15)2/2)  defined for all x.

(B)    A pdf is often used to model a random event.  For example, let us consider the following probabilistic model.  Suppose that R is the amount of rainfall (in inches) in Alphaville during one year. (R is called a random variable.)  Then the probability that a < R < b is given by the area under the pdf, p2, from x=a to x=b.

(1)   Find the probability that the amount of rainfall in Alphaville is more than 16 inches during a year.

(2)   Find the probability that the amount of rainfall in Alphaville is less than 13 inches during a year.

(C)   The mean value (or average value) of a random variable with pdf p(x) is given by the (improper) integral of xp(x) over the real line.  Find the mean value of the random variable R defined in part (B).

(D)    The variance of a random variable with pdf p(x) is given by the (improper) integral of (x-m)2p(x) over the real line, where m denotes the mean value of the random variable.   Find the variance of R.

(E)     Suppose that the amount of snowfall, S, (in inches) during one year in Betaville is given by the pdf p1(x) defined in (A).  Find the mean value and the variance of S.

III   Laplace Transforms

For a function f(x), the Laplace transform of f is defined by the improper integral:

if the integral converges.  The Laplace transform is very valuable in solving differential equations.

For each function given below, compute its Laplace transform. You may need to assume that s > 0.  Recall that in Maple we write:  assume(s>0);

(A)    f(x) = 1
(B)     f(x) = sin 3x    (You may need to use the simplify command.)
(C)    f(x) = cos 5x
(D)    f(x) = e-3x sin 4x
(E)     f(x) = eax   (Assume that a < 0.)
(F)     f(x) = (sin x) / x


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