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Statistics

Notes
Example 39: An unbiased die is tossed 600 times. Use normal approximation to binomial
to find the probability obtaining
(i) more than 125 aces,
(ii) number of aces between 80 and 110,
(iii) exactly 150 aces.

Solution.
Let X denote the number of successes, i.e., the number of aces.

1 1 5
 m = np = 600 ´ = 100 and  = npq = 600 ´ ´ = 9.1
6 6 6
(i) To make correction for continuity, we can write
P(X > 125) = P(X > 125 + 0.5)

-
æ 125.5 100ö
P
Thus, (X  125.5 ) P z= ç  ÷ = P (z  2.80 )
è 9.1 ø
-
=
0.5000 P= - (0 z£ £ 2.80 ) = 0.5000 0.4974 0.0026.
(ii) In a similar way, the probability of the number of aces between 80 and 110 is given by

-
-
æ 79.5 100 110.5 100ö
P (79.5 X£ £ 110.5 ) P= ç £ z £ ÷
è 9.1 9.1 ø
z
= P ( 2.25 z- £ £ 1.15 ) P= (0 z£ £ 2.25 ) P+ ( 0 £ £ 1.15 )
= 0.4878 + 0.3749 = 0.8627

æ 19.5 20.5ö
(iii) P(X = 120) = P(119.5 £ X £ 120.5) = P ç £ z £ ÷
è 9.1 9.1 ø
= P(2.14 £ z £ 2.25) = P(0 £ z £ 2.25) – P(0 £ z £ 2.14)

= 0.4878 – 0.4838 = 0.0040

15.9.7 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 ³ 10, the distribution of X can be taken
X m
-
as approximately normal with mean m and standard deviation m so that z = is a
m
standard normal variate.

Example 40: 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.

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