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June 20th, 2017, 02:06 PM   #1
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Find a, b, and c if µ(X) = 0, σ^2(X) = 1/15

Let X be a random variable with range [−1,1] and density function f = ax^2 + bx + c if |x| < 1 and 0 otherwise.

(d) Find a, b, and c if µ(X) = 0, σ^2(X) = 1/15, and sketch the graph of fX(x) .

for this, I thought about taking the definite integral of ∫(ax^2+bx+c)dx from -1 to 1, to get (a/3)+(b/2)+c+(a/3)+(-b/2) to eventually get ((2a)/3)+2c=0. Then, I thought about getting a by itself to get a=-3c, and for b, since they canceled, I would have to say that B may not exist. Also, as for C, I got (-a/3), how should you find a, b, and c?
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June 20th, 2017, 10:41 PM   #2
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you have 3 equations for the 3 unknowns

$f(x) = a x^2 + b x + c$

$\displaystyle \int \limits_{-1}^1 ~f(x) ~dx = 1$

$\displaystyle \int \limits_{-1}^1 ~xf(x) ~dx = 0$

$\displaystyle \int \limits_{-1}^1 ~x^2 f(x) ~dx = \dfrac{1}{15}$

these should be relatively straightforward to expand and solve.

yell back if you need further assistance.
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