Continuous Random Variables

What is a (probability) density function?

A function f : R → R is called a (probability) density function if:

• f(x) >/ 0 for all real x and

• _-∞^∞∫f(x) dx = 1

What is a continuous random variable?

A function X : Ω → R is called a continuous random variable if there is a (probability) density function f_X with P(X ∈ [a, b]) = ^a_b∫f(x) dx, -∞ \< a < b \< ∞

What is a distribution function?

The distribution function of a crv x with density function f_X is the map F_X : R → [0, 1] defined for all real t by F_X(t) := P(X \< t) = _-∞^t∫ f_X(x) dx

What are some notes about distribution function F_X?

• F_X is strictly increasing on [a, b]

• lim_t→-∞ F_X(t) = 0 & lim_t→∞ F_X(t) = 0

What is F_X’(t)?

f_X(t)

What is the mass/density function of a discrete r.v. vs a continuous r.v.?

• Discrete: p_X(k), k ∈ S_X

• Continuous: f_X(x), x is real

What is the distribution function F_X of a discrete r.v. vs a continuous r.v.?

• Discrete: _{k∈Sx, k\<t}∑ p_X(k), F_X is a step function

• Continuous: _-∞^t∫ f_X(x) dx, F_X is continuous

What is the relationship between the mass/density function and distribution function F_X of a discrete r.v. vs a continuous r.v.?

• Discrete: p_X(k) = F_X(k) - P(X < k)

• Continuous: f_X(x) = F’_X(x)

What is a continuous uniform distribution?

We say that X ~ unif[a, b] if density function is

f_X(x) = {1/(b-a), if x ∈ [a, b]

{0, if x ∉ [a, b]

In this case, the distribution function is

F_X(t) = {0, if t < a

{(t-a)/(b-a) if a \< t \< b

{1, if t > b

What is the exponential distribution?

Let λ > 0. We say a c.r.v. X follows the exponential distribution if the density function of X is,

f_X(x) = {λe^-λx, x >/ 0

{0, x < 0

We write X ~ exp_λ

In this case, the distribution function is,

F_X(t) = {1 - e^-λt, t >/ 0

{0, t < 0

Let X ~ exp_λ with λ > 0. Then for t, s >/ 0,…

P(X >/ t + s | X >/ t) = P(X >/ s)

What is the error function?

The error function Φ is the map Φ : R → [0, 1] given by Φ(t) = 1/√2π _-∞^t∫ e^-x²/2 dx, t is real

How can the value of Φ(t) for t > 0 by determined?

Φ(t) = 1 - Φ(-t), t is real

What is the normal distribution?

Let μ be real and σ > 0. A c.r.v. X follows the normal distribution if density function f_X is given by

f_X(x) = 1/(σ√2π) e^{-(x-μ)²/2σ²}, x is real

and we write X ~ N(μ, δ²)

The distribution function is

F_X(t) = Φ((t - μ)/σ), t is real

What is the standard normal distribution?

Let N ~ N(0, 1) and let μ be real and σ > 0. Then μ + σN ~ N(μ, σ²) when and μ = 0, σ² = 1, N(0, 1), this is the standard normal distribution

What does it mean for a collection of discrete or continuous r.v.s X₁, …, X_n to be independent?

If, for all sets A₁, …, A_n ⊆ R, we have P(X₁ ∈ A₁, …, X_n ∈ A_n) = _i=1ⁿΠ P(X_i ∈ A_i)