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