SOLUTION: The density function of X is f(x)={a+bx^2 if 0<=x<=1 0 otherwise the expectation of x is E(x)=3.75 find a and b

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Question 430162: The density function of X is
f(x)={a+bx^2 if 0<=x<=1
0 otherwise
the expectation of x is E(x)=3.75
find a and b
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
The area under the density curve must be 1:
int%28%28a%2Bbx%5E2%29%2C+dx%2C+0%2C1%29+=+1
The expectation is 3.75:
int%28x%28a%2Bbx%5E2%29%2C+dx%2C+0%2C1%29+=+3.75, or int%28%28ax%2Bbx%5E3%29%2C+dx%2C+0%2C1%29+=+3.75.
Upon integration of the first, we get a%2Bb%2F3+=+1, or a+=+1-b%2F3.
Upon integration of the second, we get a%2F2+%2B+b%2F4+=+3.75, or
1%2F2+-+b%2F6+%2B+b%2F4+=+3.75 after substitution.
==> b = 39
==> a = 1-39/3 = -12.