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Chapter 5. Discrete Random Variables and Probability Distributions

Unit 5.14. Distribution function

The full name from NCT is a cumulative distribution function but I am going to stick to the short name (used in more advanced texts). This is one of the topics most students don't get on the first attempt.

Exercise 5.9. Electricity consumption sharply increases when everybody starts using air conditioners, and this happens when temperature exceeds $latex 20\,^{\circ}\mathrm{C}$. The utility company would like to know the likelihood of a jump in electricity consumption tomorrow.

1. Consider the probability $latex P(T \le 15)$ that $latex T$ will not exceed $latex 15\,^{\circ}\mathrm{C}$. How does it relate to the probability $latex P(T \le 20)$?
2. Suppose in the expression $latex P(T \le t)$ the real number $latex t$ increases to $latex +\infty$. What happens to the probability?
3. Now think about $latex t$ going to $latex -\infty$. Then what happens to $latex P(T \le t)$?

Distribution function. Let $latex X$ be a random variable and $latex x$ a real number. The distribution function $latex {F_X}$ of $latex X$ is defined by $latex {F_X}(x) = P(X \le x).$

Properties

1. $latex {F_X}(x)$ is the probability of the event $latex \{ X \le x\}$, so the value $latex {F_X}(x)$ belongs to $latex [0,1]$.
2. As the event becomes wider, the probability increases. This property is called monotonicity and is formally written as follows: if $latex {x_1} \le {x_2}$, then $latex \{ X \le {x_1}\} \subset \{ X \le {x_2}\}$ and $latex {F_X}({x_1}) \le {F_X}({x_2})$.
3. As $latex x$ goes to $latex +\infty$, the event $latex { X \le x}$ approaches a sure event $latex \{X < +\infty\} = \mathbb{R}$ and $latex {F_X}(x)$ tends to 1.
4. As $latex x$ goes to $latex -\infty$, the event $latex \{ X \le x\}$ approaches an impossible event $latex \{X < -\infty\} = \varnothing$ and $latex {F_X}(x)$ tends to 0.
5. From (i)-(iv) the graph of the distribution function in Figure 5.4 emerges.

Figure 5.4. Distribution function shape.Most values of $latex X$ are concentrated where the growth of $latex {F_X}(x)$ is the fastest (say, between $latex a$ and $latex b$).

1. In many applications we are interested in probability of an event that $latex X$ takes values in an interval $latex \{ a < X \le b\}$. Such probability can be expressed in terms of the distribution function. Just apply the additivity rule to the set equation $latex \{ -\infty < X \le b\} = \{ -\infty < X \le a\} \cup \{ a < X \le b\}$ to get $latex {F_X}(b) = {F_X}(a) + P(a < X \le b)$ and, finally,

$latex P(a < X \le b) = {F_X}(b) - {F_X}(a). \qquad (\textrm{Eq. 5.16})$

Interval formula. Eq. (5.16) can be called an interval formula. In terms of Figure 5.4 this means subtracting one area from another.

1. The definition of $latex {F_X}(x)$ equally applies to discrete and continuous variables. However, in case of a discrete random variable the argument $latex x$ is often restricted to the values taken by the variable. From the working definition of a discrete random variable it should be clear that, if the values are ordered, then

$latex {F_X}({x_1}) = P(X \le {x_1}) = P(X = {x_1}) = {p_1},$

$latex {F_X}({x_2}) = P(X \le {x_2}) = P(X = {x_1}) + P(X = {x_2}) = {p_1} + {p_2},$

$latex ...$

$latex {F_X}({x_n}) = {p_1} + ... + {p_n}. \qquad (\textrm{Eq. 5.16.1})$

The sums arising in Eq. (5.16.1) are called cumulative probabilities. We summarize our findings in Table 5.10.

Table 5.10. Probabilities and cumulative probabilities.

Probabilities can be restored from the cumulative probabilities:

$latex {p_1} = {c_1},\,\,{p_2} = {c_2} - {c_1},\,...,\,{p_n} = {c_n} - {c_{n - 1}}.$

1. In Chapter 6 we shall see that a continuous random variable takes any fixed value with probability zero. Therefore for such variables $latex {F_X}(x) = P(X \le x) = P(X < x).$

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