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Various approximations for distributions are studied, especially those involving the Binomial, Poisson, gamma, and Gaussian (normal) distributions. m-procedures are used to make comparisons. A simple approximation to a continuous random variable is obtained by subdividing an interval which includes the range (the set of possible values) into small enough subintervals that the density is approximately constant over each subinterval. A point in each subinterval is selected and is assigned the probability mass in its subinterval. The combination of the selected points and the corresponding probabilities describes the distribution of an approximating simple random variable. Calculations based on this distribution approximate corresponding calculations on the continuous distribution.

Binomial, poisson, gamma, and gaussian distributions

The Poisson approximation to the binomial distribution

The following approximation is a classical one. We wish to show that for small p and sufficiently large n

P ( X = k ) = C ( n , k ) p k ( 1 - p ) n - k e - n p n p k !

Suppose p = μ / n with n large and μ / n < 1 . Then,

P ( X = k ) = C ( n , k ) ( μ / n ) k ( 1 - μ / n ) n - k = n ( n - 1 ) ( n - k + 1 ) n k 1 - μ n - k 1 - μ n n μ k k !

The first factor in the last expression is the ratio of polynomials in n of the same degree k , which must approach one as n becomes large. The second factor approaches one as n becomes large. According to a well known property of the exponential

1 - μ n n e - μ as n

The result is that for large n , P ( X = k ) e - μ μ k k ! , where μ = n p .

The Poisson and gamma distributions

Suppose Y Poisson ( λ t ) . Now X gamma ( α , λ ) iff

P ( X t ) = λ α Γ ( α ) 0 t x α - 1 e - λ x d x = 1 Γ ( α ) 0 t ( λ x ) α - 1 e - λ x d ( λ x )
= 1 Γ ( α ) 0 λ t u α - 1 e - u d u

A well known definite integral, obtained by integration by parts, is

a t n - 1 e - t d t = Γ ( n ) e - a k = 0 n - 1 a k k ! with Γ ( n ) = ( n - 1 ) !

Noting that 1 = e - a e a = e - a k = 0 a k k ! we find after some simple algebra that

1 Γ ( n ) 0 a t n - 1 e - t d t = e - a k = n a k k !

For a = λ t and α = n , we have the following equality iff X gamma ( α , λ ) .

P ( X t ) = 1 Γ ( n ) 0 λ t u n - 1 d - u d u = e - λ t k = n ( λ t ) k k !

Now

P ( Y n ) = e - λ t k = n ( λ t ) k k ! iff Y Poisson ( λ t )

The gaussian (normal) approximation

The central limit theorem, referred to in the discussion of the gaussian or normal distribution above, suggests that the binomial and Poisson distributions should be approximated by the gaussian.The number of successes in n trials has the binomial ( n , p ) distribution. This random variable may be expressed

X = i = 1 n I E i where the I E i constitute an independent class

Since the mean value of X is n p and the variance is n p q , the distribution should be approximately N ( n p , n p q ) .

A graph of the Gaussian approximation to the binomial: n=300, p=0.1. The x-axis represents the values of k ranging from 10-50, while the y-axis shows range of density from 0.01-0.08. The distribution plotted rises and falls at an equal rate with its peak at (30,0.075). The distribution occurs over a series of vertical bars with their heights roughly approximate to the corresponding position of the distribution. 'The actual distribution looks like a bell curve'. A graph of the Gaussian approximation to the binomial: n=300, p=0.1. The x-axis represents the values of k ranging from 10-50, while the y-axis shows range of density from 0.01-0.08. The distribution plotted rises and falls at an equal rate with its peak at (30,0.075). The distribution occurs over a series of vertical bars with their heights roughly approximate to the corresponding position of the distribution. 'The actual distribution looks like a bell curve'.
Gaussian approximation to the binomial.

Use of the generating function shows that the sum of independent Poisson random variables is Poisson. Now if X Poisson ( μ ) , then X may be considered the sum of n independent random variables, each Poisson ( μ / n ) . Since the mean value and the variance are both μ , it is reasonable to suppose that suppose that X is approximately N ( μ , μ ) .

It is generally best to compare distribution functions. Since the binomial and Poisson distributions are integer-valued, it turns out that the best gaussian approximaton is obtainedby making a “continuity correction.” To get an approximation to a density for an integer-valued random variable, the probability at t = k is represented by a rectangle of height p k and unit width, with k as the midpoint. Figure 1 shows a plot of the “density” and the corresponding gaussian density for n = 300 , p = 0 . 1 . It is apparent that the gaussian density is offset by approximately 1/2. To approximate the probability X k , take the area under the curve from k + 1 / 2 ; this is called the continuity correction .

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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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