Continuous Random Variable
Topics Covered- CDF , PDF of Continuous Random Variables along with Markov’s and Chebyshev’s inequality.
Prerequisite- Random Variables
As mentioned in my first article a random variable X with CDF Fₓ is said to be continuous random variable if Fₓ is continuous at every x. CDF has no jumps or steps .
Cumulative Distribution Function (CDF)
CDF of a random variable X , denoted Fₓ(x) is a function R to [0,1] defined as Fₓ(x)= P(X ≤ x)
Properties
- Fₓ(b)-Fₓ(a) = P(a ≤ X ≤ b)
- Fₓ is a non-decreasing function which takes non-negative values
- As X -> -∞ Fₓ goes to 0
- As X ->∞ as Fₓ goes to 1
Continuous Random Variable
A random variable with CDF Fₓ(x) is said to be continuous if Fₓ(x) is a continuous function.
- CDF has no steps or jumps
- So , P(X=x)=0 for all x
- And P( a ≤ X ≤ b) = P( a< X<b)
- Example of such distributions — Exponential , Gaussian Distributions
Probability Density Function
A continuous random variable X with CDF Fₓ(x) is said to have PDF fₓ(x) for all x₀
Fₓ(X₀) = ∫ fₓ(x) dx (this integral is from negative infinity to x₀)
CDF is integral of PDF
For a random variable X with PDF fₓ , an event A is a subset of the real line and its probability is computed as
Why do we study PDF?
Value of PDF around fₓ(x₀) is related to taking value around x such that higher the PDF , higher chance X lie there. So it helps in making probability computations easier.
∫ fₓ(x) = P(a ≤ X ≤ b) is always 1.
Side Note-
The set of possible outputs is called the support
supp(X)= {x : fₓ(x)>0}
Functions of Continuous Random Variable
Theorem: Monotonic differentiable function
Suppose X is a continuous random variable with PDF fₓ . Let g(x) be monotonic for x ∈ supp(X) with derivative g’(x) = dg(x)/dx . Then, the PDF of Y = g(X) is
Expected value of function of continuous random variable:
Let X be a continuous random variable with density fₓ(x). Let g : R → R be a
function. The expected value of g(X), denoted E[g(X)], is given by
whenever the above integral exists. The integral may diverge to ±∞ or may not exist in some cases.
Expected value (mean) of a continuous random variable:
Mean, denoted E[X] or μX or simply μ is given by
Variance of a continuous random variable:
Variance, denoted Var[X] or σ²ₓ or simply σ² is given by
Continuous Random Variables have 2 major Inequalities
Markov’s inequality:
If X is a continuous random variable with mean μ and non-negative supp(X) (i.e.P(X < 0) = 0), then
Chebyshev’s inequality:
If X is a continuous random variable with mean ( μ )and variance σ² , then
