lecture unit 4.3
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LECTURE UNIT 4.3. Normal Random Variables and Normal Probability Distributions. Understanding Normal Distributions is Essential for the Successful Completion of this Course. Recall: Probability Distributions p(x) for a Discrete Random Variable. p(x) = Pr(X=x) Two properties - PowerPoint PPT PresentationTRANSCRIPT
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LECTURE UNIT 4.3
Normal Random Variables and Normal Probability
Distributions
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Understanding Normal Distributions is Essential for
the Successful Completion of this Course
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Recall: Probability Distributions p(x) for a
Discrete Random Variable p(x) = Pr(X=x) Two properties
1. 0 p(x) 1 for all values of x
2. all x p(x) = 1
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Graph of p(x); x binomial n=10 p=.5; p(0)+p(1)+ … +p(10)=1
Think of p(x) as the areaof rectangle above x
p(5)=.246 is the areaof the rectangle above 5
The sum of all theareas is 1
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Recall: Continuous r. v. x
A continuous random variable can assume any value in an interval of the real line (test: no nearest neighbor to a particular value)
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Discrete rv: prob dist functionCont. rv: density function Discrete random
variable
p(x): probability distribution function for a discrete random variable x
Continuous random variable
f(x): probability density function of a continuous random variable x
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Binomial rv n=100 p=.5
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The graph of f(x) is a smooth curve
f(x)
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Graphs of probability density functions f(x)
Probability density functions come in many shapes
The shape depends on the probability distribution of the continuous random variable that the density function represents
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Graphs of probability density functions f(x)
f(x)
f(x) f(x)
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a b
Probabilities:area undergraph of f(x)
P(a < X < b) = area under the density curve between a and b.
P(X=a) = 0
P(a < x < b) = P(a < x < b)
f(x)P(a < X < b)
X
P(a X b) = f(x)dxa
b
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Properties of a probability density function f(x)
f(x)0 for all x the total area under the
graph of f(x) = 1
0 p(x) 1 p(x)=1
Think of p(x) as the areaof rectangle above x
The sum of allthe areas is 1
xx
Total areaTotal areaunder curveunder curve
=1=1
f(x)f(x)
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Important difference
1. 0 p(x) 1 for all values of x
2. all x p(x) = 1
values of p(x) for a discrete rv X are probabilities: p(x) = Pr(X=x);
1. f(x)0 for all x
2. the total area under the graph of f(x) = 1
values of f(x) are not probabilities - it is areas under the graph of f(x) that are probabilities
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Next: normal random variables