Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If X is a continuous random variable with range $[x_l,\infty)$ and p.d.f.

$f_x(X) \propto x^{-a}$, for $x\in[x_l,\infty)$

for some values $x_l > 0$ and $a \in \mathbb{R}$.

How do I calculate the range of values for a which $f_x(X)$ is a valid p.d.f.?

share|improve this question
    
Integrate, and deduce a(xl). –  gnometorule Jan 5 '12 at 15:32
    
The area under the pdf is $1$. So, integrate $f(x) = cx^{-a}$ from $x_1$ to $\infty$ and choose $c$ such that the value of the integral equals $1$. Hint: you probably need to restrict $x_1 > 0, a> 0$ for this to work. –  Dilip Sarwate Jan 5 '12 at 15:35
    
@DilipSarwate I have -c(xl)^(-a) = 1-a but how do I get the range for a which is valid? –  jamess Jan 5 '12 at 17:17
    
First, check your math! Then express $c$ in terms of $a$ and $x_1$, and decide: for fixed $x_1 > 0$, and knowing that $c$ must be positive (think why this must be so!), what is the range of values of $a$ for which your formula for $c$ gives positive values for $c$? Remember in all this that $x_1$ is a fixed positive number. Set it to $1$ if you like so as to not clutter up things. –  Dilip Sarwate Jan 5 '12 at 17:29
    
DUPLICATE: This question is nearly verbatim the same as this one: math.stackexchange.com/questions/96704/… –  Michael Hardy Jan 5 '12 at 19:30

2 Answers 2

up vote 1 down vote accepted

A probability density function $f$ must satisfy:

1) $f(x)\ge 0 $ for all $x$,

and

2) $\int_{-\infty}^\infty f(x)\, dx =1$.

Your density has the form $$f(x)=\cases{c \cdot x^{-a} &, x\ge x_l \cr 0&,\text{ otherwise}}$$ where $x_l>0$.

We need 1) to hold; $f$ must be non-negative.

When does that happen?

The first thing to note here is that, since $x_l>0$, it follows that $x^{-a}\ge0$; and thus $c$ must be positive in order for 1) to hold.

So far so good. $a$ can be any number (so far as we have surmised) and, for $c>0$, $f$ would define a density as long as condition 2) holds.

Your task now is to figure out when it does.

A hint towards achieving that end would be to consider when the integral appearing in 2) is converges. If the integral does converge, you can then select $c$ so that it converges to 1; and in this case, $f$ would indeed define a density.

If the integral does not converge, then $f$ would not define a density.

Read no further if all you want is a hint...


To determine the range of values of $a$ for which $f$ is a density we need to determine when $$\tag{3}\int_{x_l}^\infty c x^{-a}\,dx$$ converges.

Towards this end, note that the integral in (3) is convergent if and only if $a>1$. This is because the $p$-integral $\int_{x_l}^\infty {1\over x^p}\,dx $ converges if and only if $p>1$ (the lower limit presents no problems, since $x_l>0$).

This answers your question as to what range of values of $a$ (I assume $a$) give a valid density.

If you have $a>1$ and want to find the value of $c$, use 2): set $$ 1=\int_{x_l}^\infty cx^{-a}\,dx =\lim_{b\rightarrow\infty} { -cx^{-a+1}\over -a+1}\biggl|_{x_l}^b={cx_l^{1-a}\over a-1}, $$ then solve for $c$.

share|improve this answer

Say you've found $$ \int_{x_\ell}^\infty x^{-a}\;dx. $$

This is a valid pdf if the integral is finite; it is not a valid pdf if the integral is $\infty$.

The family of distributions we're dealing with here are called the Pareto distributions, after the Italian economist Vilfredo Pareto (1848--1923). It arises from Pareto's way of modeling the distribution of incomes. Pareto proposed that $$ \log N = A - a\log x $$ where $N$ is the number of people whose incomes are more than $x$. A bit of trivial algebra shows how the density arises from what Pareto proposed, but Pareto neglected to think about $x_\ell$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.