Warm-up1. A concert hall has 2000 seats. There are 1200 seats

on the main floor and 800 in the balcony. 40% of those in the balcony buy a souvenir program. 50% of those on the main floor buy a souvenir program. At a certain performance all seats are occupied. If an audience member is selected at random, what is the probability that a program was purchased?

2. A spinner on a full circle can take on any decimal value between 0 and 400. What is the probability that the spinner will land between 175 and 225?

Continuous Random VariablesDensity Functions

Continuous Random Variables

Possible values contain an entire interval.

Ex: Weight of newborns Nearest pound

Nearest tenth of pound4 5 6 7 8 9

4 5 6 7 8 9

Fit more & more rectangles

It approaches a curve as the rectangles become smaller & has greater accuracy.

Density Function

• Probability distribution for a continuous random variable (f(x)).

• The graph is a smooth curve called the density curve.

• F(x) 0• Total area under the curve = 1.

Uniform Distribution

All occur in equal distributions

Ex: .5 4 6

( )0

if xf x

otherwise

What’s the area from 4.5 to 5.5? What’s the area from 5.5 to 6?

Discrete:

(3 7) (3) (4) (5) (6) (7)

(3 7) (4) (5) (6)

P x P P P P P

P x P P P

Continuous:

(3 7) (3 7)P x P x

Why?

It’s like finding the area of a rectangle with width = 0.

Probabilities for continuous random variables are usually calculated using cumulative areas.

P(x<0.5)

Found using integrals & calculus – but we’ll use tables!

If we have a uniform continuous function from 3 to 8, find the height.

Ex.

Find P(x < 10)

Find P(x < 35)

0.02

50 minutes

Ex:

Find P(x<4)

Find P(x<2)

0.25

Ex:

Find P(x<20)

Find P(x>70)

Find P(20<x<70)

0.02

50 100

Homework

P. 365 (20-26)