[SOLVED] Math154 homework 3- random variables p0

MATH154
Homework 3
Random Variables
Problem 3.1: The Gamma distribution with shape α > 0 and rate λ > 0 has support on [0,∞). It is used in econometrics. The probability density function is
.
a) What distribution do we get in the case α = 1?
b) Verify that f satisfies the properties of a PDF.
c) Compute the expectation E[X] and variance Var[X].
d) Compute the moment generating function MX(t).
e) Why is a Gamma distributed random variable in Lp for all p? Problem 3.2: Verify that for θ > 0 the Maxwell distribution

is a PDF of a probability distribution on R+ = [0,∞). This distribution can model the speed distribution of molecules in thermal equilibrium. Now compute its expectation E[ .
Problem 3.3: Benford’s law deals with the statistics of the first significant digit in data. Simon Newcomb found the law in 1881 and Frank Benford made significant progress to understand it in 1938. The distribution appears also in naturally occurring sequences. For example, if you look at the first digit of the sequence 2n then the first significant digit k appears with probability pk = log10(1 + 1/k). The digit 1 for example occurs with about log10(2) = 0.30 which is 30 percent.
a) What is its expectation and variance of the distribution?
b) Verify that the sequence 2n produces this distribution. Probability Theory

Figure 1. The Benford distribution for the first significant digit. It is computed with
Histogram[Table[First[IntegerDigits[2n]],{n,1,10000}],10]
Problem 3.4: For a centered Cauchy distributed random variable, the probability density is ( ). As seen in class you can generate random variables with this distribution. Define X(x) = x on (Ω = R,B,P = f(x)dx).
a) Check that the random variable X is not in L1.
b) Look up the definition of convergence in the sense of Cauchy and verifythat the expectation of the distribution in this generalized sense. c) What can you say about the variance and higher moments or momentgenerating function of a Cauchy distributed random variable? d) Why again does Cot(PiRandom[]) generate Cauchy distributed random variables?
Problem 3.5: The support K of the law µ of a random variable is the largest closed subset of R such that µ((x − a,x + a)) > 0 for every x ∈ K and a > 0.
a) There are absolutely continuous distribution functions for which thesupport is a Cantor set on [0,1]. Construct one. (Note that this can not be the standard Cantor set because the Standard Cantor set has measure zero.)
b) There are singular continuous distributions for which the support is[0,1]. Construct one.
c) There are pure point distributions for which the support is [0,1]. Construct one.
d) Verify that for every closed set K in [0,1] there exists a measure
which has K as support.