I've got a question here that I've been working and I've ran into some difficulty.
"Let X be a mixed random variable with distribution function:
F(x) = 0 if x < 0
cx + 1/4 if 0 <= x < 1 (c is some constant) 1/3 if 1 <= x <2 1 if x >= 2
As for my notation above, I'm unfamiliar with LaTex, but what I'm trying to show you is a piecewise function. Here <= means "less than or equal to".
Now the question asks to find the value of c, given the expected value is 1. Here is what I have done:
First, find the values of the probabilities at 0,1 and 2
Using limits, we find P(X=0) is 1/4. Similiarly, we find P(X=1) is 1/12 -c (here I subtracted c+1/4 from 1/3, using the idea of limits again)
Finally, we find P(X=2) is 1- 1/3 = 2/3.
Now, use these values to calculate the expected value:
Here is where the confusion arises. First, I split the E[X] into three components:
1) the definite integral of d/dx [(cx + 1/4)times x] between 0 and 1
2) 1/12 - c times 1
3) 2 times 2/3
Now, I add these all up and equate:
c/2 +4/3 +1/12 -c = 1,
hence solving c = 5/6, however this leads to a negative probability for P(X=1), as 1/12 - 5/6 is less than zero.
Where have I gone wrong in my above calculations? I think it's to do with my expected value calculation.