- #1
toothpaste666
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Homework Statement
Suppose the distance X between a point target and a shot aimed at the point in a coin-operated target game is a continuous random variable with pdf
f(x) = { k(1−x^2), −1≤x≤1
0, otherwise.
(a) Find the value of k.
(b) Find the cdf of X.
(c) Compute P (−.5 < X ≤ .5).
(d) Find the expected distance between a point target and a shot aimed.
The Attempt at a Solution
a) [itex] k\int_{-1}^1(1-x^2)dx [/itex]
[itex]= k[\int_{-1}^1dx-\int_{-1}^1x^2dx] [/itex]
[itex]= k[x\Big|_{-1}^1-\frac{1}{3}x^3\Big|_{-1}^1] [/itex]
= k(2-2/3) = 1
k(4/3) = 1
k = 3/4b) [itex] \frac{3}{4} \int_{-1}^X(1-x^2)dx [/itex]c) [itex] \frac{3}{4}[x\Big|_{-.5}^{.5}-\frac{1}{3}x^3\Big|_{-.5}^{.5}] [/itex]
= (3/4)(1-(1/3)[2(1/3)(1/8)])
= (3/4)(1-1/36)
= .7292
d) [itex] \frac{3}{4}\int_{-1}^1x(1-x^2)dx [/itex]
[itex] =\frac{3}{4}\int_{-1}^1(x-x^3)dx [/itex]
[itex]= \frac{3}{4}[\int_{-1}^1xdx-\int_{-1}^1x^3dx] [/itex]
[itex]= \frac{3}{4}[\frac{1}{2}x^2\Big|_{-1}^1-\frac{1}{4}x^4\Big|_{-1}^1] [/itex]
= 0am I doing this right?