Continuous random variable

Question 1

Definition Continuous random variable

Answer 1

A random variable X, taking values in R, is said to be continuous if there exists a function

fx: R–>[0,+∞)

such that for every x ϵ R

Fx(x) = ∫ fx(t) dt

The function fx is called the density of X

Question 2

Theorem Continuous random variable

Answer 2

Let

f: R–> [0,+∞) such that

∫ f(x) dx = 1

Then there exists a random variable X such that

fx = f

Question 3

Properties of continuous random variable

Answer 3

▪ Fx is continuous. This implies
P(X=x) = 0 ∀x ϵ R

▪ If A = [a, b]
P(a<X<b) = ₐ∫ᵇ f(x) dx
Moreover
P(a<X<b) = ₐ∫ᵇ f(x) dx =
= -∞∫ᵃ f(x) dx - -∞∫ᵇ f(x) dx

Question 4

Conditions of continuous random variable

Answer 4

A sufficient condition is the following:

▪ Fx is continuous.
▪ Fx is differentiable up to a finite number of “exceptionable point”
▪ Except at the exceptional points, F’x is continuous

If these properties are satisfied then X is a continuous random variable and

    | F'x(x)     if x isn't an                                  |      |                exceptional point fx(x)|
    | any value >0           if x is an |      |                exceptional point

| exceptional point

Question 5

Expectation for continuous random variable

Answer 5

E(X) = ∫ x fx(x) dx

g: R –> R
E(g(x)) = ∫ g(x) fx(x) dx

Question 6

Uniform random variable

Answer 6

Let a<b
We are going to define random variables that correspond to
“choose at random a point in [a,b]”

A random variable X is called Uniform in [a,b] and we write

X~U(a,b) or X~Unif(a,b)

    | 0            if x !ϵ [a,b]                                  fx(x)|
    | 1/(b-a)   if x ϵ [a,b]

Question 7

Exponential random variable

Answer 7

We say that X is an exponential random variable with parameter λ>0, and we write

X~Exp(λ)

    | λ e^(-λx)             for x>0                                  fx(x)|
    | 0                         for x<0

E(X) = ∫ x fx(x) dx = ∫ x λ e^(-λx) dx
E(X) = 1/λ

Var(X) = E(X²) - (E(X))² = 1/λ²

Question 8

Normal random variable (Z)

Answer 8

A random variable Z is called a standard normal if:

fz(z) = (1/ √(2π))*e^(-z²/2)

Fz(z) = -∞∫ (1/ √(2π))*e^(-z²/2) dz = Ф(z)

Z~N(0,1)

E(Z) = 0

Var(Z) = 1

Question 9

Normal random variable (X)

Answer 9

Suppose Z is a standard normal,
µ ϵ R,
T>0,
X = TZ + µ

We have a normal random variable if:
▪ X~N( µ ; T² )
▪ fx(x) = (1/√(2πT²))exp[(1/2T²)(x-µ)²
▪ E(X) = µ
▪ Var(X) = T²

((X-µ)/T) ~ N(0,1)