Question

Let x be the amount of time (in minutes) that a particular San Francisco commuter must wait for a BART train. Suppose that the density curve is as pictured below (a uniform distribution): A horizontal line segment is graphed on the coordinate plane. The horizontal x axis is labeled "Minutes" and has two tick marks at 0 and 20. The vertical axis is labeled "Density" and has one tick mark at 0.05. The line enters the viewing window at (0, 0.05) and stops at (20, 0.05). (a) What is the probability that x is less than 8 min? more than 14 min? P (x is less than 8 minutes) = P (x is more than 14 minutes) = (b) What is the probability that x is between 7 and 11 min? P (x is between 7 and 11 minutes) = (c) Find the value c for which P(x < c) = .9. c = mins

Answer 1

a)
`P(X <8)= F(8) =(8)/(20)= 0.4`

`P(X>14) = 1-P(X<14) = 1-F(14) = 1-(14)/(20)= 0.3`

b)
`P(7< X<11)= F(11) -F(7) = (11)/(20) -(7)/(20)= 0.55-0.35=0.20`

c) We want to find a value c who satisfy this condition:

`P(x<c) = 0.9`

And using the cumulative distribution function we have this:

`P(x<c) = F(c) = (c-0)/(20-0) =0.9`

And solving for c we got:

`c = 20*0.9 = 18`

Explanation:

For this case we define the random variable X as he amount of time (in minutes) that a particular San Francisco commuter must wait for a BART train, and we know that the distribution for X is given by:

`X \sim Unif (a=0, b =20)`

Part a

We want this probability:

`P(X <8)`

And for this case we can use the cumulative distribution function given by:

`F(x) = (x-a)/(b-a) = (x-0)/(20-0)= (x)/(20)`

And using the cumulative distribution function we got:

`P(X <8)= F(8) =(8)/(20)= 0.4`

For the probability
`P(X>14)` if we use the cumulative distribution function and the complement rule we got:

`P(X>14) = 1-P(X<14) = 1-F(14) = 1-(14)/(20)= 0.3`

Part b

We want this probability:

`P(7< X<11)`

And using the cdf we got:

`P(7< X<11)= F(11) -F(7) = (11)/(20) -(7)/(20)= 0.55-0.35=0.20`

Part c

We want to find a value c who satisfy this condition:

`P(x<c) = 0.9`

And using the cumulative distribution function we have this:

`P(x<c) = F(c) = (c-0)/(20-0) =0.9`

And solving for c we got:

`c = 20*0.9 = 18`