91Ó°ÊÓ

The length of the line connecting two places is the distance between them. If the two locations are on the same horizontal or vertical line, the distance between them can be calculated by subtracting the non-overlapping coordinates.

Define random variables X and Y to be positions of the ambulance and the accident on the road. We are given that X, Y$~$Unif(0,L) and that X and Y are independent. Hence, the joint density function of X and Y is

role="math" localid="1647333868733" $$\begin{gathered} f(x,y)=F_X\left(x\right)F_Y\left(y\right) \\ =\frac{1}{L^2} \end{gathered}$$

for $(x,y)∈(0,L^2)$, otherwise it is equal to zero. Random variable that marks the distance between these two positions is Z:= (X - Y). Lets find the cumulative distribution of Z.

$$\begin{gathered} P(Z≤a)=1-P(Z>a) \\ =1-\frac{{(L-a)}^2}{L^2} \end{gathered}$$

The last equality is got when we consider the squares (0,L)²in the two dimensional plane. Event $Z>a$is equivalent to the set of points where X and Y differ more than a. And that sets in fact consisted of two rights triangles (one upper and one lower) with the length of side L - a.

Finally the density function of Z is$f_Z\left(a\right)=\frac{d}{da}P(Z≤a)=\frac{2(L-a)}{L^2}$