Expert answer:it’s a probability and statistics class homework, not handwriting it’s by computer.Example is attached.i need to get a good grade to pass the class
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mathematica_assignment.pdf
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Lecturer : Dr. Bystrik
Probability and Statistics
University of Miami Fall 2017
Final Assignment on Technology
Problem
1
Points possible 12
2
Total
12
24
Points earned
Student name:_____________________________________________________
Student ID:________________________________________________________
Please provide print-outs for your solutions, each on a separate page, both the input and the
outputs, for the full credit.
Good luck!
Problem 1
In this problem we will use N , notation, to match the Mathematica’s notation.
Note the alternative, also common, parametrization: the Gamma r, distribution is
implemented as Gamma r, 1 ) in Mathematica.
Use Mathematica commands to create the density plots and the bar charts for the
distributions below.
Do not forget to load three packages from the Mathematica kernel (for graphics, for
continuous distributions, and for discrete distributions) at the beginning of a Mathematica
session.
(a)
N 0, 1 , N 0, 10 , N 1, 1
(b)
Gamma 1 , 1
2 2
(c)
Binomial 10, 0. 10
(d)
Poisson 1
Problem 2
In this problem we will use N , notation.
A sum of Binomial n, p variables is normally distributed if n is “large”, but p and 1 − p is
not too small compare to n, commonly np ≥ 10, n 1 − p ≥ 10:
n
Y
∑ Yk
k 1
Yk
Y
Bernoulli p
Binomial n, p
as n gets large and np ≥ 10, n 1 − p ≥ 10:
d
N np, np 1 − p
Y →
Y − np
np 1 − p
Z
N 0, 1
This is a manifestation of the CLT.
However if n is “large”, but p or 1 − p is small enough for np to remain (commonly) under
10, Poisson is a more suitable approximation for such a binomial distribution:
lim
C n, x p x 1 − p
n→
n−x
where
x
exp −
x!
np
(commonly
10)
You are asked to use Mathematica to investigate how well an appropriate Poissonian
distribution approximate the given binomial distributions:
Will the quality of the approximation improve if we keep n the same and decrease p?
Will the quality of the approximation improve if we keep p the same and increase n?
Steps:
Generate the table of values for the p.d.f.
pX 0 , pX 1 , pX 2 , pX 3 , pX 4 , pX 5
for both binomial and the corresponding Poisson distributions described below.
Generate the table of values for the differences and the ratios of the corresponding binomial
and Poisson p.d.f. values. Judge the quality of the approximation by observing, how close the
differences are to 0, how close the ratios are to 1.
(a) n
(b) n
(c) n
(d) n
10, p
10, p
50, p
50, p
0. 10
0. 01
0. 10
0. 01
…
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