Continuous Random Variables
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9 Terms
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Continuous random variables
A variable that can take any value in an interval (uncountably infinite)
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Probability Density Function (PDF)
A function f(x) such that P(a ≤ X ≤ b) = ∫(from a to b) f(x)dx
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Properties of a PDF
f(x) ≥ 0
∫(from -∞ to ∞) f(x)dx = 1
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Cumulative Distribution Function (CDF)
F(x) = ∫(from -∞ to x) f(t)dt
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Relationship between PDF and CDF
f(x) = dF(x)/dx
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Probability at a Point
For continuous variables, P(X = x) = 0
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Uniform Distribution (Continuous)
f(x) = 1/(b - a) for a ≤ x ≤ b; otherwise 0
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If X ~ Uniform(0, 1), then P(0.25 < X < 0.75) = 0.5
Example
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Transformation of a Continuous Variable
If Y = g(X), the PDF of Y is f_Y (y) = f_X (g^-1 (y)) |d/dy g^-1 (y)|
Note 
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