Page 312 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Unit 15: Normal Probability Distribution

Notes
125.5 100
Thus, P X 125.5 P z P z 2.80
9.1
0.5000 P 0 z 2.80 0.5000 0.4974 0.0026.
2. In a similar way, the probability of the number of aces between 80 and 110 is given by

79.5 100 110.5 100
P 79.5 X 110.5 P z
9.1 9.1

P 2.25 z 1.15 P 0 z 2.25 P 0 z 1.15
= 0.4878 + 0.3749 = 0.8627

19.5 20.5
3. P(X = 120) = P(119.5 X 120.5) = P z
9.1 9.1
= P(2.14 z 2.25) = P(0 z 2.25) P(0 z 2.14)
= 0.4878 0.4838 = 0.0040

Self Assessment

Fill in the blanks:

15. Normal distribution can be used as an approximation to binomial distribution when n is
large and ............................ p ........................ q is very small.
16. In Normal distribution, the standard normal variate z would vary from .............to ............

15.6 Normal Approximation to Poisson Distribution

Normal distribution can also be used to approximate a Poisson distribution when its parameter
m 10. If X is a Poisson variate with mean m, then, for m 10, the distribution of X can be
X - m
taken as approximately normal with mean m and standard deviation m so that z =
m
is a standard normal variate.

Example: A random variable X follows Poisson distribution with parameter 25. Use normal
approximation to Poisson distribution to find the probability that X is greater than or equal to 30.
Solution:
P(X 30) = P(X 29.5) (after making correction for continuity).
.
295 25
.
P z P z 09
5
= 0.5000 P(0 z 0.9) = 0.5000 0.3159 = 0.1841
Fitting a Normal Curve

A normal curve is fitted to the observed data with the following objectives:
1. To provide a visual device to judge whether it is a good fit or not.
2. Use to estimate the characteristics of the population.

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