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Let $X$ be 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 $a \in \mathbb{R}$.

Assume $x_l$ = 0.5. Let K = ceil(X) or floor(X), that is X rounded (Up or down) to the nearest integer.

i. State the range of K and derive its probability mass function p(k). Note that

Pk(K=k) = Px(k - 0.5 ≤ X < k + 0.5)

ii. Demonstrate that this equation for pk satisfies the requirements for a p. m. f.

iii. Without reference to the form derived in (i), please explain why (for small $a$)

p(k) = (2^(a-1)) * (a-1) * k^(-a)

Can someone please at least explain to me what exactly I need to do for all these steps? It's quite confusing as I have not found some proper explanations with regards to pmf anywhere. I would appreciate it even more if someone could help me solve them :)

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Hint on how to show your answer is a valid pmf. $P\{X=k\}$ is obtained by integrating a nonnegative function, and so must be nonnegative. To show that $\sum_k P\{X=k\} = 1$, start with the result that $$\int_a^b + \int_b^c = \int_a^c$$ –  Dilip Sarwate Jan 7 '12 at 13:44
    
Actually, I withdraw my previous comment without deleting it. Maybe ii. is supposed to check whether i. was done correctly, that is, there were no calculus failures. Is each $p(k)$ that you have calculated in i. nonnegative? Does the sum of the series $p(0) + p(1) + \cdots$ equal $1$ where the $p(k)$ are the numbers that you have calculated in i. ?? –  Dilip Sarwate Jan 8 '12 at 0:59

2 Answers 2

up vote 1 down vote accepted

For part (iii).

From this post, you have $$ f_X(x) =2^{a-1}(a-1)x^{-a}, \ x\ge 1/2. $$ Then $$\eqalign{ p(k)&= \int_{k-{1\over2}}^{k+{1\over2}} 2^{a-1}(a-1)x^{-a}\,dx\cr &\approx \Bigl( (k+{1\over2}) - (k-{1\over2}) \Bigr) 2^{a-1}(a-1)k^{-a}\cr & = 2^{a-1}(a-1)k^{-a}.} $$

I'm not sure if this is what you're expected to do, as the integral above can be computed exactly (and this is what you do for part (i)).

Perhaps you're expected to think of the pmf as a "bar chart". The bar chart can be approximated by the graph of the density function. Each bar has width 1, is centered at $k$, and the height is $f_X(k)= 2^{a-1}(a-1)k^{-a}$.

(I'm not sure what "for small $a$" means here, $a$ must be greater than 1 in order for $f$ to actually define a density.)

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David: I share your puzzlement about the restriction *for small $a$*. A rigorous result is that, for every fixed $a\gt1$, $k^ap(k)\to2^a(a-1)$ when $k\to+\infty$. –  Did Jan 7 '12 at 15:24

$X$ is a continuous random variable

$K$ is a discrete random variable formed by rounding $X$ to the nearest integer. So for example when $0.5 \le X \lt 1.5$ then $K=1$. You can find the probability $K$ takes a particular value by integrating the density function for $X$ between suitable limits. This will give you the proability mass function.

You can show it is a probability mass function if each probability is non-negative and their sum is $1$.

Part (iii) is slightly strange with "Without reference to the form derived in (i)". You need to find an alternative approach: induction would give you the $k^{-a}$ term.

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For the first part, by suitable limits you mean integrate between 0.5 and 1.5, right? –  Sorin Cioban Jan 7 '12 at 12:17
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The distribution of X is not exponential. –  Did Jan 7 '12 at 12:24
    
@Didier: Fair enough, I have removed that statement –  Henry Jan 7 '12 at 12:39
    
@Sorin: Integrate between 0.5 and 1.5 to find $\Pr(K=1) = \Pr (0.5 \le X \lt 1.5)$. Integrate between $k-0.5$ and $k+0.5$ to find $\Pr(K=k) = \Pr( k-0.5 \le X \lt k+0.5)$. –  Henry Jan 7 '12 at 12:42
    
Thanks :) I've done that, but how can I show it's a pmf considering that my result depends on $a$? –  Sorin Cioban Jan 7 '12 at 12:46

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