Question

Two telephone calls come into a switchboard at random times in a fixed one-hour period. Assume that the calls are made independently of one another. (a) What is the probability that the calls are made in the first quarter hour

Answer 1

Answer:

0.25

Step-by-step explanation:

From the given information:

Since the times are uniform distributed over the one hour period (0,1):

Then;

$f_1(y_1) = 1$

$f_2(y_2) = 1$

So, $Y_1$ and $Y_2$ are independent; then:

$f(y_1|y_2) = f_1(y_1)f_2(y_2) \\ \\ = 1(1) = 1$

$P(Y_1\le 0.5 Y_2 \le 0.5) = \int ^{0.5}_{0} \int ^{0.5}_{0} f(y_1,y_2) dy_{2}dy_{1}$

$\implies \int ^{0.5}_{0} \int ^{0.5}_{0} (1) dy_{2}dy_{1}$

$= \dfrac{1}{4}$

= 0.25