warm-up 1. a concert hall has 2000 seats. there are 1200 seats on the main floor and 800 in the...
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Continuous Random Variables Possible values contain an entire interval.TRANSCRIPT
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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?
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Continuous Random VariablesDensity Functions
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Continuous Random Variables
Possible values contain an entire interval.
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Ex: Weight of newborns Nearest pound
Nearest tenth of pound4 5 6 7 8 9
4 5 6 7 8 9
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Fit more & more rectangles
It approaches a curve as the rectangles become smaller & has greater accuracy.
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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.
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Uniform Distribution
All occur in equal distributions
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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?
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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
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Continuous:
(3 7) (3 7)P x P x
Why?
It’s like finding the area of a rectangle with width = 0.
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Probabilities for continuous random variables are usually calculated using cumulative areas.
P(x<0.5)
Found using integrals & calculus – but we’ll use tables!
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If we have a uniform continuous function from 3 to 8, find the height.
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Ex.
Find P(x < 10)
Find P(x < 35)
0.02
50 minutes
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Ex:
Find P(x<4)
Find P(x<2)
0.25
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Ex:
Find P(x<20)
Find P(x>70)
Find P(20<x<70)
0.02
50 100
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Homework
P. 365 (20-26)