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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 = 3/4

b) [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]

= 0

am I doing this right?