Statistics Assignment: Probability, Statistics, and MATLAB functions

This assignment solution covers several key concepts in statistics. Question 1 deals with probability calculations, including conditional probability and expected return, using given probabilities for events A and B. Question 2 explores the geometric mean, comparing it to the arithmetic mean and demonstrating its relationship to the mean of logarithms. Question 3 involves probability calculations related to incoming mail. Questions 4, 5, 6 and 7 address concepts like recurrence intervals, probability density, and the use of MATLAB functions to compute variance, standard deviation, and other statistical measures. The solution provides step-by-step working and MATLAB function calls for each problem, offering a comprehensive guide to understanding and solving the assignment.

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Research Methods and Statistics 2019 Assignment 4
By (name)
Tutor
Course
Date

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Question 1
P (A) – Probability oil is struck at location A = 0.5
P (B) – probability oil is struck at location B = 0.3
P (AB) – probability that oil is struck at both A and B = 0.2
i) Probability of A & B
Since A and B are not mutually exclusive,
P (A or B) = p(A) + p(B) – p(A and B) = (0.5 + 0.3) – 0.2 = 0.6
ii) Probability of A given B
P(A|B) = P ( A∧B)
P( B) = 0.2
0.3 =2
3
iii) Return
20 × P ( A ) +20 × P ( B )−10
¿ ( 20 ×0.5 )+ ( 20× 0.3 )−10
¿ 10 million
Matlab call function
For the conditional probability in (ii)
Question 2
i)
Assuming that a > b
4 ab= ( a+b ) 2− ( a−b )2
Assuming ( a+ b ) 2 ≫ ( a−b ) 2
4 ab< ( a+b ) 2
ab< ( a+b )2
4
√ab=(a+ b)
2
Let

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m= loge a+ loge b
2 = 1
2 loge(ab)
m=loge( √ ab)
The geometric mean is,
GM = √ ab=em=e
1
2 (loge a+loge b)
Matlab commands
X = [a, b]
Geometric mean = geomean(X)
Arithmetic mean = mean(X)
M = log ( a)+log (b)
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Question 3
Probability of incoming mail sent to T=Probability of N sent to T or Probability of U sent to T
P(T)=P(TU)+P(TN)
=(0.1*(1-0.35))+(0.35*(1-0.05)
=0.3975
ii.
p(TU)= Probability of U and T= (1-0.35)*(0.1)=0.065
Question 4
i)
Let the probability of good PCs be p
p= 3
5
The probability of PCs with hardware bugs be p '
p' = 2
5
P ( all good PCs )= 3
5 × 2
4 × 1
3 = 1
10
ii)
Let the probability of good PCs be p
p= 60
100

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The probability of PCs with hardware bugs be p '
p' = 40
100
P ( all good PCs ) = 60
100 × 59
99 × 58
98 = 1711
8085
iii)
Let the probability of good PCs be p
p= 60
100
The probability of PCs with hardware bugs be p '
p'= 40
100
P ( all good PCs ) = 60
100 × 60
100 × 60
100 = 27
125
Question 5
i)
Since the probability is the inverse of the recurrence interval,
P= 1
RI
RI = 1
P = 1
0.125 =8
ii) probability of at least one flood year in a period of 7 years
The flood interval will be,
Recurrence interval= 7
1 =7
The probability will be,
p= 1
7 ≈ 0.14
Question 6
Average occurrence per year
nav=6
1 year=52 weeks

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Rate of occurrence per week
6
52
In 10 weeks the rate of occurrence=
6
52 ×10=60
52
Let r denote the number occurrence in 10 weeks
P ( r >2 ) =1− [ p ( r =0 ) + p ( r =1 ) + p (r=2) ]
P ( R=r ) = e−λ λr
r !
P ( r=0 ) = e−λ λ0
0 ! =e− λ
P ( r=1 ) = e−λ λ1
1 ! = λ e−λ
P ( r=2 ) = e−λ λ2
2 ! = λ2 1
2 e− λ
P ( r >2 )=1−[e− λ+ λ e− λ+ 1
2 λ2 e− λ
]
But λ= 60
52
P ( r >2 )=1−[e
−60
52 + 60
52 e
−60
52 + 1
2
60
52
2
e
−60
52
]=0.11065
Question 7
The probability density is f ( x )= {a−bt ,∧0 ≤ t ≤ 2
0 ,∧otherwise
i) finding a and b
f ( 0 )=0
f ( 0 )=a−0
0=a−0
a=0
Finding b,

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∫
0
2
f ( t ) dt=1
∫
0
2
( a−bt ) dt =1
1=at−b t2
2 ¿ 0 ¿ 2
(2 a− 4 b
2 )− ( 0−0 ) =1
(2 a− 4 b
2 )=1 , but a=0
−4 b
2 =1
b=−0.5
Therefore, f(t) =t/2
ii) expected values
E ( T ) =∫
0
2
t × t
2 dt=∫
0
2
t2
2 dt = t3
6 fro m 0 ¿2
E ( T )= 8
6 = 4
3
E ( T 2 )=∫
0
2
t2 × t
2 dt=∫
0
2
t3
2 dt= t4
8 fro m0 ¿ 2
E ( T 2 )= 16
8 =2
iii) variance and standard deviation
var ( T )=E (T 2 )−(E(T ))2
¿ 2− 16
9 = 2
9
standard deviation , σ= √ var ( T ) = √ 2
9 = √ 2
3
iv) probability
x−σ ≤ T ≤ x+ σ
P ¿

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∫
4 − √2
3
4 + √ 2
3
t
2 dt= t2
2 ¿ 4− √ 2
3 ¿ 4 + √ 2
3
( 4 + √ 2
12 )
2
−( 4− √ 2
12 )
2
= 4 √ 2
9
Matlab call functions
Variance = var (T)
Standard deviation = std (T)
References
DasGupta, A. (2010). Fundamentals of Probability: A First Course. Springer Science &
Business Media.
Montgomery, D. C., & Runger, G. C. (2010). Applied Statistics and Probability for
Engineers. Hoboken, NJ: John Wiley & Sons.