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CDF $F_X$ “accumulate” probability “up to” x.
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Think of CDF as running a sliding window across PMF and adding up the probabilities associated with each outcome as that window passes by.
Lecture 5 + 6 + 7: Discrete Random Variables
Cumulative Distribution Function
- Another method to describe the distribution of random variables.
- Advantage: Can be defined for any kind of random variable (discrete, continuous, and mixed).
The CDF of discrete random variable $X$:
$$
F_X(x) = P(X \leq x) = \sum_{k \leq x}p_X(k), \quad \text{for all } x \in \mathbb{R}
$$
Properties of CDF
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$F_X$ is monotonically nondecreasing: $$ \text{if } x \leq y \text{, then } F_X(x) \leq F_X(y). $$
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$F_X$ tends to 0 as $x \to -\infty$, and to 1 as $x \to \infty$.
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for all $k \in \mathbb{Z}$ $$ F_X(k) = \sum_{i = -\infty}^{k}p_X(i), $$
$$ p_X(k) = P(X \leq k) - P(X \leq k - 1) = F_X(k) - F_X(k - 1), $$
proof for (2)
By the axiom of probability, $$ \lim_{x\to+\infty}F_X(x) = 1. $$ Using complement rule, $$ P(X > x) = 1 - P(X \leq x) = 1 - F_X(x). $$ Recall $$ P(X \in \mathbb{R}) = 1. $$ Thus, $$ \lim_{x \to -\infty} 1 - F_X(x) = P(X > -\infty) = 1 \text{ (Because X is always in some real-valued range)} $$
$$ \Leftrightarrow \lim_{x \to -\infty} F_X(x) = 0. $$