Question

P [ {(x-μ)÷Ω}<= {t}]
​

Answer 1

Notation.

• The indicator function of a set S is a real-valued function defined by :

1S(x) =

1 if x ∈ S

0 if x 6∈ S

• Suppose that f : D → R is a real-valued function whose domain is an

arbitrary set D. The support of f, written supp(f), is the set of points in

D where f is nonzero

supp(f) = {x ∈ D | f(x) 6= 0}.

1 Probability Density Function and Cumulative

Distribution Function

Definition 1.1 (Probability density function). A rrv is said to be (absolutely)

continuous if there exists a real-valued function fX such that, for any subset

B ⊂ R:

P(X ∈ B) = Z

B

fX(x) dx (1)

Then fX is called the probability density function (pdf) of the random vari-

able X.

In particular, for any real numbers a and b, with a < b, letting B = [a, b],

we obtain from Equation (1) that :

P(a ≤ X ≤ b) = Z b

a

fX(x) dx (2)

Property 1.1. If X is a continuous rrv, then

• For all a ∈ R,

P(X = a) = 0 (3)

In other words, the probability that a continuous random variable takes on

any fixed value is zero.