Cumulative Distribution Function (CDF)¶
Definition¶
Also can be called "Distribution Function".
Describes the probability that a random variable \(Y\) will take a value less than or equal to a specific value \(y\), that is:
\[
F_{Y}(y) = P(Y\leq y), \ \ \ \ \text{for } \ {-\infty}<y<\infty
\]
Key Purposes¶
- Probability Calculation:
- CDF provides the probability that \(X\) will fall within a certain range.
- ie. \(P(a < X \leq b) = F(b) - F(a)\)
- Characterization of Distributions:
- CDF fully describes the probability distribution of a random variable, whether it is continuous, discrete or mixed
- Convenience:
- CDF is easier to work with than the original PDF/PMF, especially when dealing with transformations of random variables or deriving properties of distributions.
Properties¶
If \(F_{Y}(y)\) is a distribution function, then
\[
\begin{align}
&1. \ \ \ F_{Y}(-\infty) = \lim_{ y \to -\infty }F_{Y}(y) = 0 \\
&2. \ \ \ F_{Y}(\infty) = \lim_{ y \to \infty } F_{Y}(y) = 1 \\
&3. \ \ \ F_{Y}(y) \text{ is a monotonic non-decreasing function of } y. \text{\ i.e. If } y_{1} < y_{2}, \text{ then } \ F_{Y}(y_{1}) \leq F_{Y}(y_{2}).
\end{align}
\]
References¶
Random Variable