Question

# The purpose of this exercise is to investigate the claim made in

Note 7 that P() appears as an approximation

to Binomial(n; p) when n is large, p

is small and = np:" The relevant data is provided in BinomPois.csv.

(a) First suppose n is small, with n = 10, p = 0:3 and = 3.

(i) Calculate P(Y = 4) under the Binomial(10; 0:3) distribution, and also calcu-

late P(Y = 4) under the P(3) distribution. Are these calculated probabilities

similar or dierent. (3 points)

(ii) In the dataset BinPois.csv, x contains 1000 observations generated from a

Binomial(10; 0:3) distribution and y contains 1000 observations generated

from a P(3) distribution. Construct a plot with overlaid histograms for x

and y and discuss your ndings. (2 points)

(iii) Construct both a Binomial-ness Plot and Poisson-ness Plot for x and discuss

your ndings. (3 points)

(b) Now suppose n is larger, with n = 1000, p = 0:003 and = 3.

(i) Calculate P(Y = 4) under the Binomial(1000; 0:003) distribution, and com-

pare this value to P(Y = 4) calculated under the P(3) distribution from part

(a.i.) above. (2 points)

(ii) In the dataset BinPois.csv, z contains 1000 observations generated from a

Binomial(1000; 0:003). Construct a plot with overlaid histograms for z and

y and discuss your ndings. (2 points)

(iii) Construct both a Binomial-ness Plot and Poisson-ness Plot for z and discuss

your ndings. (3 points)

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