Note that fX (t) is not probability. Actually, P(X = t) = 0 for any t if X is a continuous random variable. Because {X = t} ⊂ {t − < X ≤ t} for any P(X = t) ≤ P(t − < X ≤ t) = FX (t) − FX (t − ). Hence 0 ≤ P(X = t) ≤ lim→0 [FX (t) − FX (t − )] = 0 by continuity of FX .
I
Any meaningful statement about probability must consider X lying in some interval. Probability is interpreted as the area under the density function.
Example: Logistic distribution
A random variable X with logistic distribution if FX (x) =
1 . 1+e−x
Then fX (x) =
dFX (x) e−x = dx (1 + e−x )2
P(a < X < b) = FX (b) − FX (a) Z b Z = fX (x)dx − −∞
a
Z fX (x)dx =
−∞
If ∆x is small, P(a ≤ x ≤ a + ∆x) ≈ fX (a)∆x.
b
fX (x)dx. a
Quantile and median I
Definition: Let X be a random variable with CDF FX (x). For any 0 < α < 1, an quantile of X is any number xα satisfying FX (xα ) ≥ α
and FX (xα− ) ≤ α.
The median is x0.5 . I
Quantiles defined above may not be unique. To make it unique, we usually define the quantile as xα = inf{x : FX (x) ≥ α}.