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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

Respuesta :

ajeigbeibraheem

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;

[tex]f_1(y_1) = 1[/tex]

[tex]f_2(y_2) = 1[/tex]

So, [tex]Y_1[/tex] and [tex]Y_2[/tex] are independent; then:

[tex]f(y_1|y_2) = f_1(y_1)f_2(y_2) \\ \\ = 1(1) = 1[/tex]

[tex]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}[/tex]

[tex]\implies \int ^{0.5}_{0} \int ^{0.5}_{0} (1) dy_{2}dy_{1}[/tex]

[tex]= \dfrac{1}{4}[/tex]

= 0.25

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