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Can anyone please help me with this random variable question I've stumbled across.

Recall from calculus that a function $h$ is called non-decreasing if $x \le y$ implies $h(x) \le h(y)$, for every $x, y \in \mathop{\mathrm{dom}} h$.

Q1a) Let $X$ be a continuous random variable with probability density function $f$. Prove that the probability distribution function of $X$ is non-decreasing.

I'm assuming this means show $F(x) = \int_{-\infty}^x f(y)\,dy$, is a non-decreasing function of $x$ in $\mathbb R$.

Q1b) Show that $\lim_{x\to-\infty} F(x) = 0$ and $\lim_{x\to \infty} F(x) = 1$, and explain the probabilistic meaning of these facts.

Sorry about the layout i'm not used to using this site, hope it makes sense!

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Hint for Q1a) $f \ge 0$. –  martini May 10 '12 at 9:48
    
Hint for Q1b) $F(x) = \lim_{a \to -\infty} \int_{a}^x f(y)\ dy = \lim_{a \to -\infty} (F(x) - F(a))$. –  Robert Israel May 11 '12 at 1:39
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2 Answers

up vote 2 down vote accepted

One way to do 1(b): $F(x) = \int_{-\infty}^x f(t)\ dt$ is an improper integral, which by definition of improper integral means $\lim_{a \to -\infty} \int_a^x f(t)\ dt$. Now $\int_a^x f(t)\ dt = F(x) - F(a)$, so $$ F(x) = \lim_{a \to -\infty} (F(x) - F(a)) = F(x) - \lim_{a \to -\infty} F(a)$$ and you can solve for $\lim_{a \to -\infty} F(a)$.

As for $\lim_{x \to \infty} F(x) = \lim_{x \to \infty} \int_{-\infty}^x f(t)\ dt$, that is $\int_{-\infty}^\infty f(t)\ dt$, which according to the definition of a probability density function must be $1$.

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The proof by Robert on the upper limit does depend on f approaching ∞ as x does. Then to know that the limit equals 1 relies on the definition of a probability density. That f tends to 0 as x approaches +∞ or −∞ is a required property for a probability desnity function. –  Michael Chernick May 13 '12 at 15:29
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@MichaelChernick: where do you see $f(x) \to 0$ as $x \to \infty$ written as a required property for a probability density? The only real requirements are $f \ge 0$ and $\int_{-\infty}^\infty f(x)\ dx = 1$. Some elementary texts (wanting to avoid Lebesgue integration) may require $f$ to be piecewise continuous. But I haven't seen any that required limits of $0$. –  Robert Israel May 13 '12 at 17:59
    
As I said it is required because it would be impossible for ∫f(x) dx=1 integrating over (-∞, ∞) if f does not go to 0 as x approaches ∞!!! –  Michael Chernick May 13 '12 at 20:52
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No. For example, try an $f$ whose graph consists of infinitely many triangular "bumps", the $n$'th bump (for positive integers $n$) having height $2n$ and width $2^{-n}/n$, so area $2^{-n}$. –  Robert Israel May 13 '12 at 21:17
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Or if you prefer an analytic function, try $6 \pi^{-5/2}\sum_{n=1}^\infty n e^{-n^6 (x-n)^2}$. –  Robert Israel May 13 '12 at 21:24
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Yes because the density function is never less than 0 the integral from -∞ to x cannot be larger than from -∞ to y for any y>x. So F is monotone non-decreasing. The density function f has to go to 0 as x approaches + or - ∞. So lim x→−∞ F(x)=0. The fact that lim x→∞ F(x)=1 is a requirement for f to be a probability density with F as its CDF. There is no other way to see that f integrates to 1 from the information given. All you can deduce is that it integrates to a positive constant.

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when you say 'the integral from -∞ to x cannot be larger than from -∞ to y for any y>x' do you mean ∫-∞ to x of f(y)dy < ∫-∞ to y of f(y)dy ? –  Ricky Rozay May 11 '12 at 0:24
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@Micheal, I know this sounds silly but why does the density function f HAVE to go to 0 as x approaches + or - ∞? –  Ricky Rozay May 11 '12 at 0:33
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Not at all silly. Actually $f$ doesn't have to go to $0$ as $x \to + \infty$ or $-\infty$. Fortunately, that's irrelevant to the questions being asked. –  Robert Israel May 11 '12 at 1:36
    
@Robert Israel, can you please endeavour to spread some light on what to do? –  Ricky Rozay May 11 '12 at 9:38
    
@user31024 First of all I wouldn't use y as both the upper limit of integration and the variable being integrated. y is any fixed value greater than x and I am just saying that if a function f is greater than 0 between x and y it must integrate to something positive. If f does not tend to 0 but stays positive as it approaches either +∞ or −∞ the integral would be infinite. I don't understand what Robert is saying, If f does not go to 0 as x tends to +∞ it would not be possible for lim x→∞ F(x)=1 instead it would tend to +∞ and at some point exceed 1! –  Michael Chernick May 13 '12 at 15:23
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