CHAPTER 1,2/CLO 1,2
STAT102
14. Suppose a continuous random variable x has the probability density
0<x< 1
f(x) = {k(1- x),
10,
elsewhere
(a) Find k (b) Find P(0.1 < x < 0.2) (c) P(x > 0.5) using distribution function, determine
the probabilities that (d) x is less than 0.3 (e) between 0.4 and 0.6 (f) Calculate mean and
variance for the probability density function.

a. In order for to be a proper density function, its integral over its domain must evaluate to 1:

also must be non-negative over its support, which is the case here.

I assume the instruction regarding the distribution function doesn't apply to parts (b) and (c).

(b) Integrate the density over the interval [0.1, 0.2]:

(c) Integrate the density over the interval [0.5, 1]:

The distribution function is obtained by integrating the density:

(d) Using the distribution function, we have

(e) Using again, we get

(f) The mean is

The variance is

where

so that the variance is

its 60

4 x 12 = 48

3 x 4 = 12

48 + 12 = 60

yeeeet