The Normal

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• Lesson 15
• The Normal
• Distribution

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For a truly continuous random variable, P(X
c) 0 for any value, c.
Thus, we define probabilities only on intervals.
P(X lt a)
P(X gt b)
P(a lt X lt b)
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f(x) is the probability density function, pdf.
This gives the height of the frequency curve.
Probabilities are areas under the frequency curve!
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f(x) is the probability density function, pdf.
This gives the height of the frequency curve.
Probabilities are areas under the frequency curve!
Remember this!!!
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f(x)
x
a b
P(X lt a) P(X lt a) F(a)
P(X gt b) 1 - P(X lt b) 1 - F(b)
P(a lt X lt b) P(X lt b) - P(X lt a) F(b) - F(a)
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If X follows a Normal distribution,
with parameters m and s2, we use the notation
X N(m , s2)
E(X) m
Var(X) s2
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m
m s
m - s
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A standard Normal distribution is one where m
0 and s2 1. This is denoted by Z Z
N(0 , 1)
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Table A.3 in the textbook gives
upper-tail probabilities for a standard Normal
distribution, and only for positive values of Z.
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Table C in the notebook gives cumulative probabili
ties, F(x), for a standard Normal distribution,
for 3.89 lt Z lt 3.89.
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P(Z lt 1.27)
.8980
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P(Z lt -0.43)
.3336
P(Z gt 0.43)
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P(Z gt -0.22)
1 P(Z lt -0.22)
1 .4129 .5871
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P(-1.32 lt Z lt 0.16)
P(Z lt 0.16) - P(Z lt -1.32)
.5636 .0934 .4702
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Find c, so that P(Z lt c) .0505
c -1.64
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Find c, so that P(Z lt c) ? .9
c ? 1.28
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Find c, so that P(Z gt c) .166
c 0.97
? P(Z lt c) 1 - .166 .834
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ZP is the point along the N(0,1)
distribution that has cumulative probability p.
Z.0505 -1.64
Z.9 ? 1.28
Z.975 1.96