Mathematics · lesson 2 of 2

Continuous distributions and densities

A density is integrated over a set; its value at a point is not the point probability.

University8 practice questionsNotes + worked examples

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Learn the ideas

Prerequisites: Integration and probability.

Density versus probability

A density is integrated over a set; its value at a point is not the point probability.

Normalization

A density must be nonnegative and integrate to one.

Expected value

E[g(X)]=∫g(x)f(x)dx when the integral exists.

Cumulative distribution

F(x)=P(X≤x); differentiating an absolutely continuous CDF recovers a density almost everywhere.

Worked examples

Example 1

Let f(x)=2x for 0≤x≤1 and zero otherwise. Find P(X≤1/2).

Integrate 2x from 0 to 1/2 to obtain 1/4.

0.25

Example 2

For a continuous random variable with density f(x)=2x on [0,1] and zero elsewhere, find E[X].

Integrate 2x² from 0 to 1.

2/3

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 Let f(x)=2x for 0≤x≤1 and zero otherwise. Find P(X≤1/2).

Show answer and explanation

0.25

Integrate 2x from 0 to 1/2 to obtain 1/4.

2 For a continuous random variable with density f(x)=2x on [0,1] and zero elsewhere, find E[X].

Show answer and explanation

2/3

Integrate 2x² from 0 to 1.

3 For an absolutely continuous X, what is P(X=a) at a fixed point?

A. 1 for every aB. 0C. f(a) alwaysD. Undefined

Show answer and explanation

A single point has zero integral under a density.

4 A uniform variable on [2,8] has what mean?

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The mean of a uniform interval is its midpoint.

5 Explain how a density can exceed 1 without violating probability rules.

Sample answer and marking guidance

A density has units inverse to x; probability is area. For example a uniform density of 2 over [0,1/2] integrates to 1.

Credit a valid normalized example and area rather than height.

6 For an exponential model with rate 2, find the mean.

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0.5

The mean is 1/rate.

7 Which proposed function cannot be a probability density?

A. A function with value 3 at some pointsB. A function with a cornerC. A function negative on an intervalD. A function that is zero outside a finite interval

Show answer and explanation

A function negative on an interval

Densities must be nonnegative almost everywhere.

8 Find the constant c making f(x)=c x² on [0,1] a density.

Sample answer and marking guidance

c=3, because the integral is c/3 and must equal 1.

Credit nonnegativity and normalization.

Learn the ideas

Density versus probability

Normalization

Expected value

Cumulative distribution

Worked examples

Let f(x)=2x for 0≤x≤1 and zero otherwise. Find P(X≤1/2).

For a continuous random variable with density f(x)=2x on [0,1] and zero elsewhere, find E[X].

Check your understanding

Further reading and review references