# Continuous distributions and densities

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

Course: [Probability models and continuous distributions](https://studybrix.com/en/courses/university-probability/index.md)
Level: university
Prerequisites: Integration and probability.

## Notes

### 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.

## Practice questions

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

Answer: 0.25

Explanation: 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].

Answer: 2/3

Explanation: 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 a
B. 0
C. f(a) always
D. Undefined

Answer: 0

Explanation: A single point has zero integral under a density.

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

Answer: 5

Explanation: The mean of a uniform interval is its midpoint.

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

Sample answer / 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.

Explanation: Credit a valid normalized example and area rather than height.

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

Answer: 0.5

Explanation: The mean is 1/rate.

### 7. Which proposed function cannot be a probability density?

A. A function with value 3 at some points
B. A function with a corner
C. A function negative on an interval
D. A function that is zero outside a finite interval

Answer: A function negative on an interval

Explanation: Densities must be nonnegative almost everywhere.

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

Sample answer / marking guidance: c=3, because the integral is c/3 and must equal 1.

Explanation: Credit nonnegativity and normalization.

## Further reading

- https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/
- https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.