Continuous random variables Flashcards
Defining probability density of a continuous random variable
The function f(x) is called probability density function (PDF), it can not take negative values and area under curve y = f(x) is equal to 1.
P(1.3 < X < 3.5) = P(1.3 ≤ X < 3.5) = P(1.3 < X ≤ 3.5) = P(1.3 ≤ X ≤ 3.5) …
characteristic of continuous distributions
Notation:
f(x) = {0.2
0
for 0 ≤ x < 5, otherwise.
Properties of a probability density function
∫∞ -∞ f(x) dx = 1
P(a ≤ X ≤ b) = ∫ᵇₐ f(x) dx
Median and other percentile of a continuous random variable
Median
P(X ≤ M) = ∫ᴹ₀ f(x) dx = 1/2
Lower Quartile, Upper Quartile
P(X ≤ LQ) = ∫ᴸᵠ₀ f(x) dx = 1/4
P(X ≤ UQ) = ∫ᵁᵠ₀ f(x) dx = 3/4
IQR = UQ - LQ
Expectation and variance of a continuous random variable
E(X) = ∫∞ -∞ x f(x) dx
Var(X) = ∫∞ -∞ x² f(x) dx - µ²
(∞) upper limit if given
(-∞) lower limit if given