MAT 1235 Calculus II Section 8.5 Probability

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Presentation transcript:

MAT 1235 Calculus II Section 8.5 Probability

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)=1/6 The 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

Probability Density Function Properties:

Example 4 (a) Show that f(x) is a pdf of some random variable 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.

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

Example 4 (c) Let X be the lifetime of a type of battery (in years). Find the average lifetime of such type of batteries.

Exponential Distribution Used to model waiting times, equipment failure times. It have a parameter c. The average value is 1/c. So, c =

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.

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