Question

1. A radioactive particle is randomly located in a square area with sides that are 1 unit in length. Let X and Y denote the coordinates of the particle. Because the particle is equally likely to fall in any subarea of fixed size, a reasonable model of (X , Y ) is given by

a. Find the mean of X.

b. Find the mean of Y.

c. Find the covariance between X and Y.

d. Find the correlation between X and Y.

Answer 1

Solution:

$\begin{array}{ll} f(x, y)=\left\{\begin{array}{ll} 1, & & 0 \leq x \leq 1,0 \leq y \leq 1 \\ 0, & \text { elsewhere } \end{array}\right. \\ & f_{1}\left({x}\right)=1 \\ & f_{2}\left(y\right)=1 \end{array}$

The expected value is the sum of the product of each possibility with its probability:

(a) $E\left(X\right)=E\left(Y\right)=\int_{-\infty}^{+\infty} x f(x) d x=\int_{0}^{1} x(1) d x=\frac{1}{2}$

(b) $E(Y)=\frac12$

(c) Then we get,

$E\left(XY\right)=E\left(X\right) E\left(Y\right)=\frac{1}{2} \times \frac{1}{2}=\frac{1}{4}$

$Cov(XY)=E(XY)-E(X)E(Y)=\dfrac14-\dfrac12.\dfrac12=0$

(d) $Correlation(XY)=\dfrac{Cov(XY)}{\sigma_X\sigma_Y}=0$