Question

An article suggests the uniform distribution on the interval (7.5, 19) as a model for depth (cm) of the bioturbation layer in sediment in a certain region.

(a) What are the mean and variance of depth? (Round your variance to two decimal places.) mean variance


(b) What is the cdf of depth? F(x) = 0 x < 7.5 7.5 ≤ x < 19 1 19 ≤ x

Answer 1

a) Mean: 13.25

Variance: 11.02

b)

0 if x below than 7.5

`{x-a}{b-a}` is x in between 7.5 and 19

1 if x higher than 19.

Explanation:

This is a problem in which we use the uniform probability distribution.

(a) What are the mean and variance of depth?

The mean is the midpoint of 7.5 and 19. So

`M = (7.5 + 19)/(2) = 13.25`

The mean is 13.25

The variance is given by the following formula:

`V = ((b-a)^(2))/(12)`

In which b and a are the limits of the interval. So
`b = 19, a = 7.5`

So

`V = ((b-a)^(2))/(12) = ((19-7.5)^(2))/(12) = 11.02`

The variance is 11.02.

(b) What is the cdf of depth?

The uniform probability distribution has a cummulative distribution function, of 0 if is below a,
`(x-a)/(b-a)` for x in between a and b and 1 is x is greater than b.

So:

0 if x below than 7.5

`{x-a}{b-a}` is x in between 7.5 and 19

1 if x higher than 19.