2.1.1 Probability Functions (PMFs and PDFs) Flashcards

(8 cards)

1
Q

What is a probability density function (PDF)?

A

A function that describes the relative likelihood of a continuous random variable taking on a given value.

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2
Q

Why can’t a PDF provide exact probabilities for specific values?

A

Because a continuous random variable can take an infinite number of values, making the probability of any single value equal to zero.

The area under a vertical line is zero. DUH

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3
Q

How do you find probabilities using a PDF?

A

By integrating the PDF over a range:
P(a ≤ X ≤ b) = ∫_{a}^{b} f(x) dx.

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4
Q

What are the two conditions a valid PDF must satisfy?

A
  1. f(x) ≥ 0 for all x.
  2. The total area under the PDF must equal 1: ∫_{-∞}^{∞} f(x) dx = 1.
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5
Q

Can a PDF be greater than 1?

A

Yes, as long as its integral over all possible values equals 1.

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6
Q

What does evaluating a PDF at a specific value tell us?

A

The relative likelihood of that value occurring, not the probability.

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7
Q

What does it mean if f(x) = 0 for some values of x?

A

Those values are impossible for the random variable.

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8
Q

Example: If X represents annual rainfall and has a range of [0, 100], what is P(X = 40)?

A

P(X = 40) = 0 because the probability of any exact value for a continuous variable is zero.

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