AP Calculus BC
__________________
Probability Density Functions Name:
1-13-16
A probability density function is a function f having as its domain the set
conditions:
1.
f (x)  0 for all x °
2.



of real numbers and satisfies the following two
f (x)dx  1
*If f is a probability density function for a particular event occurring, then the probability that the event will occur over the closed
b
interval [a, b] is P([a, b]) =  f (x)dx .
a
*The mean or expected value of a probability density function f is defined to be   E(x) 
*The median of a probability density function f is the number m such that
1.
2.


m
f (x)dx 



xf (x)dx .
1
.
2
0
x 0
A common probability density function is the exponential density function defined by f (x)   kx
ke
x 0

(a)
Verify that an exponential density function is indeed a probability density function.
(b)
Verify that the mean of an exponential density function is  
where k > 0.
1
.
k
For a particular kind of battery the probability density function for x hours to be the life of the battery selected at random is
0
x 0

given by f (x)   1  x
.
60
x 0
 e
 60
3.
4.
(a)
Find the probability of the life of a battery selected at random will be between 15 and 25 hours.
(b)
Find the probability of the life of a battery selected at random will be less than 10 hours.
(c)
Find the probability of the life of a battery selected at random will be more than 50 hours.
(d)
Find the mean battery life of a battery selected at random.
In a certain city, the probability density function for x minutes to be the length of a telephone call selected at random is
0
x 0

given by f (x)   1  x
.
3
x 0
 e
3
(a)
Find the probability that a telephone call selected at random will last between 1 and 2 minutes.
(b)
Find the probability that a telephone call selected at random will last at least 5 minutes.
(c)
Find the median length of a telephone call selected at random.
0

 1
A uniform probability distribution is defined by f (x)  
d  c
0
density function.
x c
c  x  d where c  d . Show that f (x) is a probability
d x
5.
6.
A normal distribution is defined by f (x) 
1
e (x   ) / (2 ) , where  denotes the mean and  the standard deviation. IQ
2
2
 2
scores are distributed normally with a mean 100 and standard deviation 15.
(a)
What percentage of the population has an IQ score between 85 and 115?
(b)
What percentage of the population has an IQ above 140?
(a)
Explain why the function f (x) is a probability density function.
(b)
(c)
Use the graph to find:
i.
P(X  3) .
ii.
P(3  X  8) .
Calculate the mean of f (x) .