Download Section 5.4 - TopCatMath

Math 140 - Cooley
Business Calculus
OCC
Section 5.4 – Probability
Definition
Let x be a continuous random variable. A function f is said to be a probability density function for x if:
1. For all x in the domain of f, we have 0  f ( x ) .
2. The area under the graph of f is 1.
3. For any subinterval [c, d ] in the domain of f, the probability that x will be in that subinterval is given by
d
P([c, d ])   f ( x )dx .
c
Definition
A continuous random variable x is said to be uniformly distributed over an interval [a, b] if it has a probability
density function f given by
1
, a  xb.
f ( x) 
ba
Definition
A continuous random variable is exponentially distributed if it has a probability density function of the form
f ( x)  ke kx , over the interval [0, ) .
 Exercises:
Verify Property 2 of the definition of a probability density function over the given interval.
1)
f ( x)  2 x ,
2)
1
f ( x)  x 2 ,
3
[0,1]
[2,1]
-1-
Math 140 - Cooley
Business Calculus
OCC
Section 5.4 – Probability
 Exercises:
Find k such that each function is a probability density function over the given interval. Then write the probability
density function.
3)
f ( x )  kx ,
[1,4]
4)
f ( x)  ke x ,
[0,3]
5)
A number is selected at random from the interval [5,29] . The probability density function for x is
given by
1
, for 5  x  29 .
f ( x) 
24
Find the probability that a number selected is in the subinterval [14,29] .
6)
A telephone company determines that the duration t, in minutes, of a phone call is an exponentially
distributed random variable with a probability density function
f (t )  2e2t , for 0  x   .
a)
Find the probability that a phone call will last no more than 2 minutes.
b)
Find the probability that a phone call will last more than 5 minutes.
-2-