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Unit 15: Exponential Distribution and Normal Distribution

(i) The area between the ordinates at m -  and m +  is 0.6826. This implies that for a Notes
normal distribution about 68% of the observations will lie between m -  and m + .
(ii) The area between the ordinates at m - 2 and m + 2 is 0.9544. This implies that for a
normal distribution about 95% of the observations will lie between m - 2 and
m + 2.

(iii) The area between the ordinates at m - 3 and m + 3 is 0.9974. This implies that for a
normal distribution about 99% of the observations will lie between m - 3 and
m + 3. This result shows that, practically, the range of the distribution is 6s although,
theoretically, the range is from - ¥ to ¥.

15.3.5 Probability of Normal Variate in an interval

Let X be a normal variate distributed with mean m and standard deviation , also written in
abbreviated form as X – N(m, ) The probability of X lying in the interval (X , X ) is given by
1 2
2
1 X m- ö
X 2 1 - ç æ ÷
P (X £ X £ X 2 ò e 2 è  ø dX
) =
1
X 1  2p
In terms of figure, this probability is equal to the area under the normal curve between the
ordinates at X = X and X = X respectively.
1 2
Figure 15.4

Note It may be recalled that the probability that a continuous random variable takes
a particular value is defined to be zero even though the event is not impossible.

It is obvious from the above that, to find P(X £ X £ X ), we have to evaluate an integral which
1 2
might be cumbersome and time consuming task. Fortunately, an alternative procedure is available
-
X m
for performing this task. To devise this procedure, we define a new variable z = .

-
æ X m ö 1
( ) m-
E
E
We note that ( ) z = ç ÷ = [E X ] 0=
è  ø 
-
æ X m ö 1 1
and Var z ç è  ÷ ø =  2 Var (X m- ) =  2 Var ( ) 1.X =
( ) Var=
Further, from the reproductive property, it follows that the distribution of z is also normal.
Thus, we conclude that if X is a normal variate with mean m and standard deviation s, then
X m
-
z = is a normal variate with mean zero and standard deviation unity. Since the parameters


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