Suppose a random variable X follows the binomial distributio

Suppose a random variable X follows the binomial distribution with N=75 and p=0.05

c) How well you believe that the Pisson approximates the binomial result? Would the agreement have been better or worse had the original value for p equalled 0.1?

Solution

a)

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    5      
x2 =    7      
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    75      
p = the probability of a success =    0.05      
          
Then          
          
P(at most    4   ) =    0.678851637
P(at most    7   ) =    0.966371999
          
Thus,          
          
P(between x1 and x2) =    0.287520362   [ANSWER]

b)

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    5      
x2 =    7      
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    3.75      
          
          
Then          
          
P(at most    4   ) =    0.677547636
P(at most    7   ) =    0.962378658
          
Thus,          
          
P(between x1 and x2) =    0.284831021   [ANSWER]

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c)

Yes, I think it approximates the binomial result, as the difference is only around 0.003. It will become worse if p = 0.1, as Poisson approximation is better as p becomes smaller.