Suppose a student takes the bus to University every day. There are 3 stops along

Question

Suppose a student takes the bus to University every day. There are 3 stops along the way, and the probability of stopping at each of them is 0.40, 0.20 and 0.60, respectively. Assume independence of stopping at either of the three stops.

Suppose stopping at each stop adds exactly 3, 2, and 4 minutes, respectively, to the bus travel time. If no stops happen, the time to University is 12 minutes.

(a) In how many ways could the bus have to stop en route and what are the probabilities for each?

(b) Write down the probability mass function of the commute time (in minutes) to University and compute its expected value and standard deviation.

(c) Let X be the random variable representing the number of stops made en route. Write down the probability mass function of X.

(d) Find the expected number of stops made en route. Interpret this value in the context of the question.

Explanation / Answer

Thus,

Number of ways = 8

Expected number of stops = 1.2 stops.

Thus, on an average, the bus stops at 1.2 stops for large number of iterations.

Hope this helps. Ask if you have doubts.

3 min 2 min 4 min Time Ways Stop 1 Stop 2 Stop 3 Total Probability 1 0 0 0 12 0.192 2 0 1 0 14 0.048 3 1 0 0 15 0.128 4 0 0 1 16 0.288 5 1 1 0 17 0.032 6 0 1 1 18 0.072 7 1 0 1 19 0.192 8 1 1 1 21 0.048 No. of Stops Probability Product 0 0.192 0 1 0.464 0.464 2 0.296 0.592 3 0.048 0.144 Sum 1.2

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