Continuous random variables Flashcards

1
Q

Defining probability density of a continuous random variable

A

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.

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2
Q

Properties of a probability density function

A

∫∞ -∞ f(x) dx = 1

P(a ≤ X ≤ b) = ∫ᵇₐ f(x) dx

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3
Q

Median and other percentile of a continuous random variable

A

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

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4
Q

Expectation and variance of a continuous random variable

A

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

(∞) upper limit if given
(-∞) lower limit if given

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