A branch of a certain bank in New York City has six ATMs....

A branch of a certain bank in New York City has six ATMs. Let X represent the number of ATMs in use at a particular time of day. The cdf of X is as follows: F(x) = 0, if x < 0 0.06, if 0 ? x < 1 0.19, if 1 ? x < 2 0.39, if 2 ? x < 3 0.67, if 3 ? x < 4 0.92, if 4 ? x < 5 0.97, if 5 ? x < 6 1, if 6 ? x Calculate the following probabilities directly from the cdf: (a) p(2), that is, P(X = 2) (b) P(X > 3) (c) P(2 ? X ? 5) (d) P(2 < X < 5) (e) P(2 ? X < 5) (f) P(2 < X ? 5)

Transcript

00:01 So we're given the distribution f of x for variable x, which is the number of atms in use at a particular time of day, and x is either 0, 1, 2, 3, 4, 5, or 6.

00:25 That is the domain of x.

00:28 So if we want to calculate the probability that x is equal to 2, that is going to be the probability that x is less than or equal to 2, minus the probability that x is less than 2, because we only want equal.

01:00 Well, and this probability, because it's discrete, is going to be the probability that x is less than or equal to 1.

01:12 So really then, this is f of 2 minus f of 1, which is equal to 0 .39 minus 0 .19, so this is equal to 0 .20.

01:31 And then in part b, we want a probability that x is greater than 3.

01:41 Well, that's going to be the probability, nope, that's going to be 1 minus the probability that x is less than or equal to 3, because greater than 3 is 4, 5, and 6.

01:58 Less than or equal to 3 is the other options.

02:03 So then this is 1 minus f of 3, which is 1 minus 0 .67, and that then is equal to 0 .33.

02:19 For c, we want a probability that 2 is less than or equal to x, less than or equal to 5.

02:30 So, if we look at our domain here, we want 2, we want 3, we want 4, and we want 5.

02:41 So i'm going to say this is equal to the probability that x is less than or equal to 5, minus the probability that x is less than 2, which is really the probability that x is less than or equal to 1, because we have a discrete variable that only can be equal to integers.

03:05 So then this is equal to f5 minus f1, which is, f5 is 0 .97, f1 is 0 .19.

03:25 So this probability then is equal to 0 .78...

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