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Probability Distribution Function

Corollary 6.1. In the case of DRV a graph of integral function of distribution is
represented in a staircase form. 2

Definition 6.2. A random variable is called continuous if its integral function
of distribution is continuous for all x.

Now we consider the properties of an integral function of distribution.

1. 0 ≤ F(x) ≤ 1;

2. F(−∞) = lim F(x) = lim P(X < x) = 0, F(−∞) = lim F(x) = lim P(X <
x→−∞ x→−∞ x→−∞ x→−∞
x) = 0.

3. An integral distribution function is a non-decreasing function of x.

PROOF. Let x 1 < x 2 then P(X < x 2 ) = P(X < x 1 ) + P(x 1 ≤ X < x 2 ), but F(x 2 ) =
P(X < x 2 ), F(x 1 ) = P(X < x 1 ), hence

F(x 2 ) − F(x 1 ) = P(x 1 ≤ X < x 2 ). (6.2)

A probability of any event is non-negative, so F(x 1 ) ≤ F(x 2 ). Rewrite the equality (6.2) in such a form
to have
P(x 1 ≤ X < x 2 ) = F(x 2 ) − F(x 1 ). (6.3)

Thus, the probability of the occurrence of a random variable in a given interval is equal to the increment
of its distribution function on this interval. 2

4. The probability of occurrence of a continuous random variable in a specific point is equal to
zero.

PROOF. On setting in the formula (6.3) x 2 = x 1 + ∆x we get P(x 1 ≤ X < x 2 ) = F(x 1 +
∆x) − F(x 1 ). Let ∆x → 0 then

lim P(x 1 ≤ x < x 2 ) = lim (F(x 1 + ∆x) − F(x 1 )) = lim ∆F(x 1 ) = 0
∆x→0 ∆x→0 ∆x→0

since a function F(x) is continuous. 2

Corollary 6.2. The following equalities are valid for a continuous random variable

P(x 1 ≤ X < x 2 ) = P(x 1 < X < x 2 ) = P(x 1 < X ≤ x 2 ) = P(x 1 ≤ X ≤ x 2 ).

Probability Distribution Function

Definition 6.3. By probability distribution function (probability density function

or probability mass function) we mean a derivative of an integral distribution
function, that is
′
f(x) = F (x). (6.4)
✓

A density f(x) satisfies the following conditions:

1. f(x) ≥ 0 since f(x) = F (x) and F(x) is a non-decreasing function.
′

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