Question

A uniformly distributed random variable has minimum and maximum values of 20 and 60, respectively.

a. Draw the density function.
b. Determine P(35 < X < 45).
c. Draw the density function including the calculation of the probability in part (b).

Answer 1

a. The density function of a uniformly distributed random variable is a rectangle with height 1/(maximum value - minimum value) and width equal to the range of the variable. In this case, the height is 1/(60 - 20) = 1/40 and the width is 60 - 20 = 40. Therefore, the density function is:

```
|
|
|
|
| ___________
| | |
| | |
|_____|___________|
20 35 60
```

b. To find P(35 < X < 45), we need to find the area of the rectangle between x = 35 and x = 45. The height of the rectangle is 1/40 and the width is 45 - 35 = 10. Therefore, the area is:

P(35 < X < 45) = (1/40) * 10 = 1/4

c. Here is the density function with the shaded area representing the probability from part (b):

```
|
|
|
|
| ___________
| | |
| | |
|_____|___________|
20 35 60

|
|
|
|
| ___________
| | |
| | |
|_____|___________|
35 45
```

The shaded area represents the probability P(35 < X < 45), which is 1/4.