Continuous random variables

Question 1

Defining probability density of a continuous random variable

Answer 1

X can be a random continuous variable, random variable because X depends on chance, continuous variable if the variable can take any value within a given interval

Note:
The total area under the curve must be 1 since the probabilities must sum to 1

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.

Question 2

Properties of a probability density function

Answer 2

f(x) ≥ 0 or ∫∞ -∞ f(x) dx = 1

P(a ≤ X ≤ b) = ∫ᵇₐ f(x) dx

Question 3

Median and other percentile of a continuous random variable

Answer 3

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 = 3/2
IQR = UQ - LQ

Question 4

Expectation and variance of a continuous random variable

Answer 4

E(X) = ∫∞ -∞ x f(x) dx
Var(X) = ∫∞ -∞ x² f(x) dx - µ²