Continuous Random Variables

I.  Homework

Do not try problem 35 from 9.2.

II.  Definition of a Probability Density Function

Let f(x) be a continuous function on [a,b] then f(x) is a probability density function if the following holds:

1)  f(x) > 0 for all x in [a,b]

2)  

III. Examples

Let f(x) = .2 for x between 3 and 8, then

1)  f(x) > 0 and

2)  int f(x) dx = .2x|from 3 to 8 = .2(8) - .2(3) = 1.6 - 6 = 1

Hence f(x) is a pdf.

Exercises:  Determine which of  the following are probability density functions:

A)  f(x) = (4 - x)/8 on [0,4]

B)  f(x) = 24x on [-2,4]

C)  f(x) = 1/3 e^-x/3 on [0,infinity]

D)  f(x) = 1/4 xe^-x/2 [0,infinity]

IV.  Finding a Probability

Suppose that f(x) is a probability density function, then we define the probability that x lies in the interval [c,d] to be

IV.  Applications

A:  Suppose that you throw a dart on a square dart board of width 6, in such a way that the probability of the dart falling x units from the y-axis uniformly distributed (the likelihood of hitting any area of the board is the same for all areas.)  

Find the Probability Density function and determine P(-2 < x < 2) where x = 0 corresponds to the center of the board.

Solution:
Since the probability is uniform, the pdf is a constant, k. To find k, set

1 = int from -3 to 3 of kdx = kx| from -3 to 3 = 6k, hence k = 1/6.

So f(x) = 1/6

Now P(-2 < x < 2) = int from -2 to 2 of 1/6dx = 1/6x| from -2 to 2 = 2/3.

Exercises:

A.  You own a snowboard rental shop and have determined that the distribution of the amount of time a customer has to wait is f(x) = .125e^-x/8

i)  Show f is a probability density function   

ii)  Determine the probability that a customer will wait less than 5 minutes.

iii)  Determine the probability that a customer will wait more than 10 minutes.

iv)  95% of the customers will have to wait fewer than how many minutes?

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