1. Continuous random variables and probability distributions

2. Continuous random variables
Can take on any value in a given interval (or a union of intervals)
Examples: height, weight, length, mass, temperature, concentration
They are continuous in theory but in practice we measure their values to the nearest inch, gram, milligram, etc.
For continuous rv’s, there is no positive probability at a point.

3. Examples
In the study of the ecology of a lake, depth measurements are taken at randomly chosen locations. Then the depth X at a location is a continuous rv.
Let X be the amount of time that a random customer spends waiting for a haircut. Then X is neither discrete not continuous, since there is a positive probability that X=0.

4. Lake depth example
If we “discretize” lake depth and draw a histogram, so that the area of the rectangle above any possible integer k is the proportion of the lake whose depth is (to the nearest meter) k, then the total area under the rectangles is one.

5. Lake depth example (continue)
If we measure depth more accurately, we obtain a histogram with a finer mesh.
In the limit (as the mesh size gets smaller) we obtain a smooth curve, the probability density function.
The area under the curve is one.

6. Probability density function (pdf)
pdf defines the distribution of a continuous rv
pdf f(x) has to be nonnegative:
f(x) can be greater than 1 but the integral (probability) is between 0 and 1
Total area under the curve y = f(x) is one, and area under the curve for a specific interval is the probability for that interval.

7. Probability of a particular value
Thus, we don’t need to worry about < or when we talk about probabilities for a continuous random variable.

8. Example f(x)=2x, =0 otherwise Calculate P(1/4<=X<=2/3)
f(x) is nonnegative for 0<=x<=1
integrates to one
Calculate P(1/4<=X<=2/3)

9. Uniform distribution
A continuous rv X is said to have a uniform distribution on the interval [A,B] if the pdf of X is

10. Application of the uniform distribution
Consider a reference line from the stem of a tire to an imperfection on a tire, and let X be the angle (in degrees) measured clockwise to the location of an imperfection. Then