Question

The lengths of a professor's classes has a continuous uniform distribution between 50.0 min and 52.0 min. If one such class is randomly selected, find the probability that the class length is less than 50.4 min. P(X < 50.4) =

Answer 1

The probability that a randomly selected class is less than 50.4 minutes long is 20%, calculated based on the area under the continuous uniform distribution curve from 50.0 to 50.4 minutes.

Step-by-step explanation:

The question deals with the concept of continuous uniform distribution, which is a part of probability theory in mathematics. In this distribution, every outcome in a range is equally likely to occur, and the probability of an event is determined by the area under the uniform distribution curve between two points.

To find the probability that the class length is less than 50.4 minutes (P(X < 50.4)), we calculate the area under the curve from the lower limit of 50.0 minutes to 50.4 minutes. Since the distribution is continuous and uniform between 50.0 and 52.0 minutes, the density function is 1 divided by the length of the interval (52.0 - 50.0 minutes).

The calculation for the probability is therefore as follows:

P(X < 50.4) = (50.4 - 50.0) / (52.0 - 50.0)

P(X < 50.4) = 0.4 / 2

P(X < 50.4) = 0.2 or 20%

The probability that a randomly selected class is less than 50.4 minutes long is 20%.

Answer 2

`P(X<50.4)`

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

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

And if we use this formula we got:

`P(X<50.4)= F(50.4) = (50.4- 50)/(52-50)= 0.2`

And then the probability that the class length is less than 50.4 min is 0.2

Step-by-step explanation:

Let X the random variable who represent the lengths of a professor's classes, and the distribution for X is given by:

`X \sim Unif (a= 50, b=52)`

And we want to find the following probability:

`P(X<50.4)`

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

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

And if we use this formula we got:

`P(X<50.4)= F(50.4) = (50.4- 50)/(52-50)= 0.2`

And then the probability that the class length is less than 50.4 min is 0.2