# Calculating probability mass functions with constraints from cumulative distribution

This is a self-study question. The name of the book is called: Applied Statistics and Probability for Engineers by Montgomery and Runger. This problem is on page 73. It's exercise 3-41.

The entire problem is listed as the following:

Given the following cumulative distribution function: $$F(x)=\left\{\begin{matrix} 0 & & x<-10 \\ 0.25 & & -10\leq x< 30 \\ 0.75 & &30\leq x< 50 \\ 1 & & 50 \leq x \end{matrix}\right.$$ Determine each of the probabilities: a) $P(X<50)$ b) $P(0 \leq X < 10)$ c) $P(-10 < X <10)$... etc etc. The question I have is this:

Why does the following probability mass function evaluate to 0? $$P(0 \leq X < 10) = 0$$

Isn't this set of outcomes a subset of $-10\leq x< 30$ and therefore should be evaluated to 0.25?

• Where are you getting this from? Is this an example problem from a text or course? Can you provide some context? – gung Dec 19 '14 at 23:36
• Yes this is an example problem from a text book. The lesson is on deriving cdfs from pmfs and converting pmts to cdfs. There isn't any context beyond what I listed. – J.W. Dec 19 '14 at 23:39
• It would help to get the name of the book, page number, etc., you can copy & paste the full question & the surrounding descriptive text, etc. Also, please add the [self-study] tag & read its wiki. – gung Dec 19 '14 at 23:42
• Ok! I added all the context I'm able to provide. – J.W. Dec 19 '14 at 23:46
• For goodness sake, just draw $F$ (why aren't you always doing this?). Mark the two bounds on the open interval on your plot. How much does $F$ change inside that interval? I'd think this is an absolute minimum requirement for a reasonable attempt at the question. [On the other hand, if you have done at least that much, given the requirements on self-study questions, you should show your attempt.] – Glen_b Dec 20 '14 at 1:12

The cumulative probability distribution function $F_X(x)$ tells us how much probability mass there is to the left of $x$ or at $x$ for each $x$ on the real line. (The choice of notation, though almost universally used is truly dreadful for use in a classroom setting! How on earth does one read out aloud $F_X(x)$ or $P\{X\leq x\}$? F-sub-big X of little x? probability that random variable $X$ is no larger than lower-case x?) Formally, the value of $F_X(x)$ is just $P\{X \leq x\}$. As Glen_b's comment says, you really should start by sketching the function $F_X(x)$ at the very least.

When $X$ is a discrete random variable taking on values $x_1, x_2, \ldots$ with probabilities $p_1, p_2, \ldots$ respectively, a little thought (instead of rote memorization of the definition) reveals that $F_X(x)$ must be what can be described as a staircase function, increasing from $0$ to $1$ as $x$ increases, with steps of heights $p_1, p_2, \ldots$ at points $x_1, x_2, \ldots$ etc. The function is discontinuous at each $x_i$, and is constant in each interval $[x_i, x_{i+1})$ (please be sure to note the $[$ and $)$ in the description of the intervals). Note that $F_X(x_i)$ includes $p_i$ so that the value of $F_X(x)$ at the point $x=x_i$ (where the function is discontinuous) is the value on the right. Since you are studying from a text intended for engineers, you might find this written as $F_X(x) = F_X(x^+)$. Thus, $$F_X(x) = P\{X \leq x\} = F_X(x^+) ~ \text{and} ~ P\{X < x\} = F_X(x^-).$$

In fact, for any random variable (not necessarily a discrete random variable or an integer-valued random variable as in Rusan's answer) and for any real numbers $a$ and $b$ such that $a \leq b$, \begin{align} P\{a < X \leq b\} &= F_X(b^+) - F_X(a^+) = F_X(b)-F_X(a),\tag{1}\\ P\{a \leq X \leq b\} &= F_X(b^+) - F_X(a^-) = F_X(b) - F_X(a^-),\tag{2}\\ P\{a \leq X < b\} &= F_X(b^-) - F_X(a^-) = F_X(b^-) - F_X(a^-),\tag{3}\\ P\{a < X < b\} &= F_X(b^-) - F_X(a^+) = F_X(b^-)-F_X(a).\tag{4} \end{align}

For the special case when $b = a$, $(2)$ above becomes $$P\{X=a\} = F_X(a^+)-F_X(a^-),$$ that is, $P\{X=a\}$ is the jump (if any) in the value of $F_X(x)$ at $x=a$. If $F_X(x)$ is continuous at $x=a$, then $P\{X=a\}=0$.

With this as prologue, note that your given $F_X(x)$ is a staircase function with jumps of $\frac 14, \frac 12, \frac 14$ at $x=-10, 30, 50$ respectively; that is, $X$ takes on values $-10, 30, 50$ with probabilities $\frac 14, \frac 12, \frac 14$ respectively, and once you have that, the answers to the questions asked are easy to compute directly, or, if you prefer to read the $F_X(0^-)$ etc off the graph that you have drawn as you apply $(1)$-$(4)$, that is fine too.

Since this question relates specifically to a discrete random variable (as your book says...), I will answer it as such. Dilip Sarwate's answer provides the general result for a discrete random variable.

With those health warnings, the cumulative distribution function (CDF), $F$, between $x=-10$ and $x=30$ does not vary. This implies the probability mass strictly between these points is $0$. In particular, the interval $0\leq x <10$ lies strictly within the interval $-10\leq x <30$ so $\mathbb{P}(0\leq X<10)=0$.

To see why this is, assume that the CFD does change at some $x$. Because $F$ is a non-decreasing function, and $X$ is a discrete random variable, we can find a sufficiently small $\epsilon>0$ so that $$F(x-\epsilon)<F(x)$$ holds, and so can be rearranged as $$0<F(x)-F(x-\epsilon)=\mathbb{P}(X\leq x)-\mathbb{P}(X\leq x-\epsilon).$$ Because we are free to choose $\epsilon$ as small as we like, subject to $0<\epsilon$, we rewrite the inequality as $$0<\mathbb{P}(X\leq x)-\mathbb{P}(X<x)=f(x)=\mathbb{P}(X=x).$$ So the points for which the CDF changes (steps) are those that have positive probability mass. By reversing the argument, you can see the converse is also true.

• Thanks Rusan. Can you explain this step or provide a better definition that led you to this inference?: "This implies the probability mass between these points is 0." Yes, this book is quite sparse on clear explanations. – J.W. Dec 20 '14 at 2:38
• @J.W. I have added more detail. – Rusan Kax Dec 20 '14 at 12:35
• The expression $$\mathbb{P}(a\leq X \leq b)=\mathbb{P}(X\leq b)-\mathbb{P}(X<a)=F(b)-F(a-1)$$ is applicable only for discrete integer-valued random variables. – Dilip Sarwate Dec 20 '14 at 14:18
• (1) Out of curiosity, what specifically do you think is "quite poor" about this notation? (2) Referring to random variables that are "integer valued" is a little misleading (and leads to some incorrect expressions in the answer). How would your analysis change if (say) "$30$" in the question were changed to $10\pi$? – whuber Dec 20 '14 at 19:33
• @whuber Since you asked. (1) I was referring to the book, in general. (2) I accept I have not answered in full generality. The OP asked a specific question (a discrete RV taking integer values) about a specific topic in a named text. I was simply making as much of the answer "explicit" for the benefit of the OP. I will delete the answer as soon as I get back to my desk. – Rusan Kax Dec 20 '14 at 23:15