Probability Function

Question

The probability density function f(x) of a continuous random variable X is:

 

f(x)={ -kx, If -1<x<o

      ={k, If o≤x<1, where k is a positive constant.

      ={o, otherwise.

I, Graph f

II, Find the value of k

III, Compute P(-(1/2)<X< (1/2))

Answer 1

For a function to be a PDF, its definite integral over is domain must = 1.

 

The indefinite integral for the interval [-1,0) is (-k/2)x² and the associated definite integral value is k/2.

The definite integral for the interval [0,1] is k.

 

So we have k +k/2 = 1 which gives k = 2/3.

 

you can do iii.