10.Normal Distribution

NORMAL DISTRIBUTION

Normal distributions are one type of continuous probability distribution.

If X has the Normal distribution with mean µ and variance s ², this is denoted by

In order to obtain probabilities for the Normal distribution (i.e. areas under the curve), it is necessary to express any value of X in terms of the number of standard deviation units it is away from µ.

Standard Normal Distribution

Since X is a random variable so is Z. Using the formulas for functions of random variables you can obtain

Also it can be shown that Z has a Normal distribution.

Thus Z ~ N(0,1).

This is called the standard Normal distribution.

Probabilities for Normal distributions other than the standard Normal distribution N(0,1) are obtained by using the formula

to convert from X ~ N (µ, s ²) to Z ~ N(0,1) and then using the table of probabilities for N(0,1).

Example - Within 1 standard deviation of the mean

You can use MINITAB to obtain these probabilities

e.g. If X ~ N(5,9) (i) find P(X < 7) (ii) find k such that P(X < k) = 0.05

(i)  CDF  7;
     NORMAL   5   3.          gives prob = 0.7475

(ii) INVCDF   0.05;
     NORMAL   5   3.          gives k = 0.0654
     (Check these using tables)

Example - Soft drink bottle filler example revisited

Recall the example about soft drinks.

A filling machine is used to fill soft drink bottles.

The bottles are supposed to contain 300 mls.

In fact the quantities vary according to the Normal distribution with expected value of µ = 302 ml and standard deviation s = 3ml.

What is the probability that an individual bottle contains less than 295 mls?

Let the random variable X denote the quantity in an individual bottle. We are told X ~ N(302, 3²), and we want Pr{X < 295}.

If   X = 295   then   Z = (295 - 302)/3 = -2.33

so P(X < 295) = P(Z < -2.33)

= 1 - P(Z < +2.33)

= 1 - .990

= 0.01

i.e. about 1 bottle in 100 would have less than 295 ml.