Question

State the properties of distribution function.

Answer 1

The function F_x(x) or simply F(x) has the following properties.

(i) 0 ≤ F(x) ≤ 1, `-∞` < x < `∞`

(ii) `"F"(-∞) = lim_(x -> - oo) "F"(x) = 0  "and"  "F"(+ oo) =  lim_(x -> oo)  "F"(x)` = 1.

(iii) F(.) is a monotone, non-decreasing function; that is, F(a) < F(b) for a < b.

(iv) F(.) is continuous from the right; that is, `lim_("h" -> 0)  "F"(x + "h")` = F(x).

(v) F(x) = `"d"/("d"x) "F"(x) = "f"(x) ≥ 0`

(vi) F'(x) = `"d"/("d"x) "F"(x)` = f(x) ⇒ dF(x) = f(x)dx

dF(x) is known as probability differential of X.

(vii) P(a ≤ x ≤ b) = `int_"a"^"b" "f"(x)  "d"x = int_(-oo)^"b" "f"(x)  "d"x - int_(-oo)^"a" "f"(x) "d"x`

= P(X ≤ b) – P(X ≤ a)

= F(b) – F(a)