MAT 1235Calculus II

Section 8.5

Probability

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HW

WebAssign 8.5 (6 problems, 65 min.) Quiz: 8.2, 8.5

Preview

Provide a 30-minute snapshot of probability theory and its relationship with integration.

Preview

Provide a 30-minute snapshot of probability theory and its relationship with integration.

Engineering: MAT2200 (3) Math major/minor: MAT 3360 (5)

Random Variables

Variables related to random behaviors

Example 1

Y=outcome of rolling a die

=

X=lifetime of a Dell computer

=

Q: What is a fundamental difference between X and Y?

Continuous Random Variables

Take range over an interval of real numbers.

Probability…

of an event = the chance that the event will

happen

Example 2

P(Y=1)=1/6The chance of getting “1” is ___________

P(3≤X≤4)The chance that the Dell computer breaks

down____________________

Probability…

…of an event = the chance that the event

will happen

…is always between 0 and 1.

Example 3

P(Y=7)=

P(0 ≤ X<)=

Probability Density Function

Continuous random variable X The pdf f(x) of X is defined as

The prob. info is “encoded” into the pdf

b

a

P a b f xX x d

Probability Density Function

Properties:

1. 0 for all

2. 1

f x x

f x dx

Example 4

(a) Show that f(x) is a pdf of some random variable X.

2 1

4 12 if 02

0 Otherwise

x x xf x

Example 4

(b) Let X be the lifetime of a type of battery (in years). Find the probability that a randomly selected sample battery will last more than ¼ year.

2 1

4 12 if 02

0 Otherwise

x x xf x

130.8125

16

Average Value of a pdf

Also called 1. Mean of the pdf f(x) 2. Expected value X

xf x dx

Example 4

(c) Let X be the lifetime of a type of battery (in years). Find the average lifetime of such type of batteries.

2 1

4 12 if 02

0 Otherwise

x x xf x

xf x dx

17

0.354 years48

Exponential Distribution

Used to model waiting times, equipment failure times. It have a parameter c.

The average value is 1/c. So, c =

0 if 0

if 0ct

tf t

ce t

Example 5

The customer service at AT&T has an average waiting time of 2 minutes.

Assume we can use the exponential distribution to model the waiting time. Find the probability that customer will be served within 5 minutes.

Example 5

Let T be the waiting time of a customer.

0 if 0

if 0ct

tf t

ce t

0.92

Remarks

If a random variable is not given, be sure to define it.