the normal distribution. bell-shaped density the normal random variable has the famous bell-shaped...
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The Normal Distribution
Bell-shaped Density
• The normal random variable has the famous bell-shaped distribution. The most commonly used continuous distribution.
• The normal distribution is used to approximate other distributions (see Central Limit Theorem).
• For a normal distribution with E(Y) = and V(Y) = .
2 2( ) /(2 )1( )
2yf y e
Standard Normal Curve
• For the standard normal distribution E(Y) = 0 and V(Y) = 1.
2 / 21( )
2yf y e
Normal Probabilities
• For finding probabilities, we compute
2 2( ) /(2 )1( )
2
b y
aP a Y b e dy
• Or, at least approximate the value numerically using an algorithm like Simpson’s Rule (Calc. 1?)
Calculator Syntax
• For a normal distribution with E(Y) = and V(Y) = to compute P( a < Y < b ): normalcdf( a, b, )
Normal Probabilities
• For a < , compute P( Y < a ): 0.5 – normalcdf( a, , )
• Hence, P( y < 7.5 ) = 0.5 – 0.39435 = 0.10565
Normal Probabilities
• For b > , compute P( Y < b ): 0.5 + normalcdf( , b, )
• Hence, P( y < 11.8 ) = 0.5 + 0.31594 = 0.81594
Bearing diameters
• Quality control requires bearings to measure 3.00 + 0.004 inches in diameter.
• Currently, the bearings produced are normally distributed with mean 3.001 inches and standard deviation 0.002 inch.
• What percentage of bearings currently being produced fail to meet the given tolerance?
On a “historical” note…
• Given , , and a value x , we may determine a corresponding value z
• Such that P( < Y < x) is the same as the probability P( < Y < z) for the standard normal distribution.
• Consider P(10 < Y < 14.5) where = 10 and
14.5 102.25
2z
Transformed to Standard• Compare P( < Y < 14.5) with = 10 and
to P( < Y < 2.25) for the standard normal distribution.
0.0122
Allows you to use Table 4 in the text:
Backwards?
• For a standard normal distribution, find z such that P( Z < z ) = 0.8686
• For a normal distribution with = 5 and = 1.5, find b such that P( Y < b ) = 0.8686
• If a soft drink machine fills 16-ounce cups with an average of 15.5 ounces, what is the standard deviation given that the cup overflows 1.5% of the time?
Practice Problems
• For homework, practice
4.49, 4.57, 4.59, 4.61, 4.63
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