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) =  .

Standard Normal Curve For the standard normal distribution E(Y) = 0 and V(Y) = 1.

Normal Probabilities For finding probabilities, we compute 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 – =

Normal Probabilities For b > , compute P( Y < b ): normalcdf( , b,  ) Hence, P( y < 11.8 ) = =

Bearing diameters Quality control requires bearings to measure inches in diameter. Currently, the bearings produced are normally distributed with mean inches and standard deviation 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 

Transformed to Standard Compare P(  < Y < 14.5) with  = 10 and  to P(  < Y < 2.25) for the standard normal distribution Allows you to use Table 4 in the text:

Backwards? For a standard normal distribution, find z such that P( Z < z ) = For a normal distribution with  = 5 and  = 1.5, find b such that P( Y < b ) = 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