(4). Suppose that it has been observed that, on average, 180 cars per hour pass

Question

(4). Suppose that it has been observed that, on average, 180 cars per hour pass a specified int on a particular road in the morning rush hour. Due to impending roadworks it is estimated that congestion will occur closer to the city center if more than 4 cars pass the point in any one minute. What is the probability of congestion occurring? (5). At a raffle, 1000 tickets are sold at $2 each for four prices $500, $250, $150 and $75 Suppose you buy one ticket. Assuming the random variable X represents the value of your gain (Amount won- cost of the ticket): (a). Construct the probability mass function for the random variable X in tabular form. (b). Calculate the expected value of your gain.

Explanation / Answer

Question 4. Expected number of cars shall pass through the point = 180/60 = 3 cars per minute

so we can say it is a poisson disribution. = 3 cars per minute

Pr(X >4) = 1 - Pr(X <= 4) = 1 - POISSON (X <=4 ; 3) = 1 - 0.8153 = 0.1847

Question 5.

Number of tickets = 1000

Ticket Price = $2

There are four gains $500, $250, $150, $75

so X = the gain

pmf(x) = 1/1000 ; X = $ 498

= 1/1000 ; X = $ 248

= 1/1000 ; X = $ 148

= 1/1000 ; X = $ 73

= 996/1000 ; X = - $2

(b) Expected Value of the gain = 1/1000 [498 + 248 + 148 + 73] + 996/1000 * (-2) = - $ 1.025

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