Question

One person is driving a car along the street and passing 3 cross-sections with traffic lights. The lights are only showing red or green. Let X represent the number of cross-sections he passed before seeing the first red-light. Show the p.d.f. of X.

Answer 1

X represents the number of cross-sections he passed before seeing the first red-light

Then X can take the following values

• X = 0 if the first cross-section has a red light
• X = 1 if the second cross-section has a red light
• X = 2 if the third cross-section has a red light
• X = 3 if no cross-section has red light

Now X = 0 has probability P(X=0) = 1/2

X = 1 has probability P(X=1) = 1/2 * 1/2 = 1/4

X=2 has probability P(X=2) = 1/2*1/2*1/2 = 1/8

X = 3 has probability P(X=3) = 1/2*1/2*1/2 = 1/8

Hence the probability distribution becomes:

X	P(X=x)
0	1/2
1	1/4
2	1/8
3	1/8

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