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Notes › EENG 3421: Probability and Statistics for Engineers Lecture 9

Cumulative Distribution Function

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Cumulative Distribution Function

Definition 09.1 (Cumulative Distribution Function).

Let $X$ be a real valued random variable The cumulative distribution function of $X$ is

$$ F_X(x) = P[X \le x] $$
Proposition 09.2 (CDF Determines Interval Probabilities).

For any real numbers $a < b$

$$ P[a < X \le b] = F_X(b) - F_X(a) $$

Proof. Write the event $\{X \le b\}$ as a disjoint union

$$ \{X \le b\} = \{X \le a\} \cup \{a < X \le b\} $$

Take probabilities and subtract $P[X \le a]$

Corollary 09.3 (One Sided Probability From CDF).
$$ P[X > a] = 1 - F_X(a) $$
Corollary 09.4 (Closed Interval Probability).
$$ P[a \le X \le b] = F_X(b) - \lim_{x \uparrow a} F_X(x) $$

Basic Properties

Theorem 09.5 (Basic Properties of a CDF).

For any CDF $F_X$

  • Boundedness
$$ 0 \le F_X(x) \le 1 $$
  • Monotonicity
$$ x_1 \le x_2 \implies F_X(x_1) \le F_X(x_2) $$
  • Right continuity
$$ \lim_{h \downarrow 0} F_X(x+h) = F_X(x) $$
  • Limits at infinity
$$ \lim_{x \to -\infty} F_X(x) = 0 $$$$ \lim_{x \to \infty} F_X(x) = 1 $$
Proposition 09.6 (Jump Size Equals Point Mass).

For any real number $x$

$$ P[X=x] = F_X(x) - \lim_{t \uparrow x} F_X(t) $$

Discrete Random Variables

Proposition 09.7 (CDF in Terms of the PMF).

If $X$ is discrete with PMF $P_X$

$$ F_X(x) = \sum_{t \in S_X \,:\, t \le x} P_X(t) $$
Corollary 09.8 (PMF in Terms of the CDF).

If $X$ is discrete then for any $x \in S_X$

$$ P_X(x) = P[X=x] = F_X(x) - \lim_{t \uparrow x} F_X(t) $$
Corollary 09.9 (Probability of a Set From the CDF).

For any set $B \subseteq S_X$

$$ P[X \in B] = \sum_{x \in B} \bigl(F_X(x) - \lim_{t \uparrow x} F_X(t)\bigr) $$

References

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