Continuous random variables: the pdf
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Flip to reveal answersWhat two conditions make f(x) a valid probability density function?
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All 8 Flashcards — Continuous random variables: the pdf
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Question
What two conditions make f(x) a valid probability density function?
Answer
f(x) ≥ 0 everywhere, and the total area ∫ over all x of f(x) dx = 1.
Question
For a continuous random variable, how do you find P(a < X < b)?
Answer
Integrate the pdf: P(a<X<b) = ∫ from a to b of f(x) dx (the area under the curve).
Question
Why is P(X = a) = 0 for a continuous variable?
Answer
A single point has zero width, so zero area; probability comes from intervals (areas).
Question
How do you find an unknown constant k in a pdf?
Answer
Integrate f over its support, set the result equal to 1, and solve for k.
Question
f(x) = kx for 0 ≤ x ≤ 4. Find k.
Answer
∫₀⁴ kx dx = 8k = 1, so k = 1/8.
Question
f(x) = kx² for 0 ≤ x ≤ 3. Find k.
Answer
∫₀³ kx² dx = 9k = 1, so k = 1/9.
Question
Can a pdf take values greater than 1?
Answer
Yes — it's a density, not a probability. Only the total AREA must equal 1.
Question
Does P(a < X < b) differ from P(a ≤ X ≤ b) for a continuous variable?
Answer
No — endpoints have probability 0, so < and ≤ give the same area.
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Full study notes for Continuous random variables: the pdf
Topic 4.14 hub
Continuous random variables (HL only)
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