Mahmoud Abas, Full-Stack Software Engineer & AI

Modeling and Simulation (2026-2027) - Input Modeling: Estimation, Testing and Correlation

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Modeling and Simulation - Lesson 4

Input Modeling: Estimation, Testing and Correlation

Estimate parameters from data, test a candidate distribution and measure dependence: Exercise 4 solved step by step.

Objectives

Estimate the sample mean and variance from a frequency table
Check whether two jointly discrete random variables are independent
Estimate the parameters of lognormal data from natural logarithms
Use the chi-square test to check whether data is Poisson distributed
Compute E, Var, Cov and Cor for jointly continuous variables
Measure the correlation between two recorded variables

Part 1

What this exercise does

Six questions, four jobs

Four jobs in six questions

Job	Questions
Estimate parameters from data	Q1, Q3
Test a candidate distribution	Q4
Check dependence	Q2, Q5
Measure correlation	Q5, Q6

Part 2

Sample mean and variance

Q1: arrivals recorded for 100 days

Question 1

Q1: arrivals over 100 days

At a particular time of day, for a fixed time (10 minutes for example), arrivals are recorded
The record covers 100 days, classified by arrivals per day
Estimate the sample mean and variance

Question 1

Q1: values 0 to 4

m	Frequency f	f × m	f × m²
0	12	0	0
1	10	10	10
2	19	38	76
3	17	51	153
4	10	40	160

Question 1

Q1: values 5 to 9

m	Frequency f	f × m	f × m²
5	8	40	200
6	7	42	252
7	5	35	245
8	5	40	320
9	3	27	243

Question 1

Q1: values 10 and 11, and the totals

m	Frequency f	f × m	f × m²
10	3	30	300
11	1	11	121
Total	100	364	2080

Question 1

Q1: the sample mean

X̄ = ∑cj=1 f_j m_jn

∑ f_j m_j = (0 × 12) + (1 × 10)

+ ... + (11 × 1) = 364

X̄ = 364100 = 3.64

Question 1

Q1: the sample variance

S² = ∑cj=1 f_j m_j² - n X̄²n - 1

S² = 2080 - 100 (3.64)²100 - 1

S² = 2080 - 1324.9699

S² = 7.63

A slip on the exercise sheet

Part 3

Dependence of discrete variables

Q2: a joint probability function

Question 2

Q2: are X and Y independent?

p(x, y) = x + y30

Valid for x = 0, 1, 2
and for y = 0, 1, 2, 3
Otherwise p(x, y) = 0

P_X(x) P_Y(y) = p(x, y)

Question 2

Q2: the distribution of X, the sum over y

P_X(x) = ∑3y=0 x + y30

= x30 + x + 130

+ x + 230 + x + 330

= 4x + 630 = 2x + 315

Question 2

Q2: the distribution of Y, the sum over x

P_Y(y) = ∑2x=0 x + y30

= y30 + y + 130 + y + 230

= 3y + 330 = y + 110

Question 2

Q2: the product

P_X(x) P_Y(y) = (2x + 3)(y + 1)15 × 10

= 2xy + 3y + 2x + 3150 ≠ p(x, y)

Question 2

Q2: three points

Point	p(x, y)	Product	Match
(0, 0)	0 / 150	3 / 150	No
(1, 2)	15 / 150	15 / 150	Yes
(2, 3)	25 / 150	28 / 150	No

Part 4

Parameters of lognormal data

Q3: the rate of return on 10 investments

Question 3

Q3: 10 investments

The rate of return on the investments:
18.8, 27.9, 21, 6.1, 37.4
5, 22.9, 1, 3.1, 8.3
The data is lognormal: estimate its parameters

Question 3

Q3: the natural logarithms

x	ln x	x	ln x
18.8	2.9	5	1.6
27.9	3.3	22.9	3.1
21	3	1	0
6.1	1.8	3.1	1.1
37.4	3.6	8.3	2.1

Question 3

Q3: the estimated mean

2.9 + 3.3 + 3 + 1.8 + 3.6

+ 1.6 + 3.1 + 0 + 1.1 + 2.1

= 22.5

μ = 22.510 = 2.25

Question 3

Q3: the estimated variance

σ² = ∑10i=1 x_i² - 10 μ²10 - 1

∑10i=1 x_i² = 62.29

σ² = 62.29 - 10 (2.25)²9

= 11.6659 = 1.30

A slip on the exercise sheet

62.29 - 52.99 = 1.04

Part 5

Testing a distribution: chi-square

Q4: is the arrival data Poisson?

Question 4

Q4: the setup

The data of Q1: n = 100 days
H0: the data is Poisson distributed
The parameter is estimated by the mean: α = 3.64

Question 4

Q4: the expected frequency

f(x) = e^-α α^xx!

E_i = n f(x)

f(x) applies at x = 0, 1, 2, ...
Otherwise f(x) = 0
E is the frequency expected if H0 is true

Question 4

Q4: x = 0 and x = 1

f(0) = e^-3.64 = 0.0263

E = 100 × 0.0263 = 2.6

f(1) = 3.64 e^-3.64 = 0.0956

E = 100 × 0.0956 = 9.6

Q4: the two values

A slip on the exercise sheet

Question 4

Q4: x from 0 to 4

x	Observed O	f(x)	Expected E
0	12	0.0263	2.6
1	10	0.0956	9.6
2	19	0.1739	17.4
3	17	0.2110	21.1
4	10	0.1920	19.2

Question 4

Q4: x from 5 to 9

x	Observed O	f(x)	Expected E
5	8	0.1398	14.0
6	7	0.0848	8.5
7	5	0.0441	4.4
8	5	0.0201	2.0
9	3	0.0081	0.8

Question 4

Q4: x = 10 and x = 11

x	Observed O	f(x)	Expected E
10	3	0.0030	0.3
11	1	0.0010	0.1

Question 4

Q4: combining small intervals

Interval x	O	E
0, 1	12 + 10 = 22	2.6 + 9.6 = 12.2
7 to 11	17	7.6

Question 4

Q4: terms 1 to 5

Interval x	O	E	(O - E)² / E
0, 1	22	12.2	7.87
2	19	17.4	0.15
3	17	21.1	0.80
4	10	19.2	4.41
5	8	14.0	2.57

Question 4

Q4: terms 6 and 7

Interval x	O	E	(O - E)² / E
6	7	8.5	0.26
7 to 11	17	7.6	11.63

(22 - 12.2)²12.2 = 7.87

Question 4

Q4: the statistic

χ₀² = ∑7i=1 (O_i - E_i)²E_i

χ₀² = 7.87 + 0.15 + 0.80

+ 4.41 + 2.57 + 0.26 + 11.63

= 27.69

Question 4

Q4: degrees of freedom

k = 7: the class intervals after combining
s = 1: the number of estimated parameters
α = 0.05: the level of significance

k - s - 1 = 7 - 1 - 1 = 5

χ²_{0.05, 5} = 11.1

Question 4

Q4: reading the table

The full table is on the lesson page

ν	χ² 0.025	χ² 0.05	χ² 0.10
3	9.35	7.81	6.25
4	11.14	9.49	7.78
5	12.8	11.1	9.2
6	14.4	12.6	10.6
7	16.0	14.1	12.0

Question 4

Q4: the decision

χ₀² = 27.69 > χ²_{0.05, 5} = 11.1

Q4: the statistic

A slip on the exercise sheet

(7 - 8.5)²8.5 = 0.26

88.367.6 = 11.63

The chi-square table

A slip on the exercise sheet

Part 6

Jointly continuous variables

Q5: dependence, covariance and correlation

Question 5

Q5: the joint function

f(x, y) = y - x

Valid for 0 ≤ x ≤ 1 and 1 ≤ y ≤ 2; otherwise f(x, y) = 0
(a) Check the dependence of X and Y
(b) Compute E(X), Var(X), E(Y), Var(Y), Cov(X, Y) and Cor(X, Y)

Question 5

Q5 (a): the function of X alone

f_X(x) = ∫21 f(x, y) dy

= ∫21 (y - x) dy

= [ y²2 - xy ]_y=1²

f_X(x) = 32 - x

0 ≤ x ≤ 1

Question 5

Q5 (a): the function of Y alone

f_Y(y) = ∫10 f(x, y) dx

= ∫10 (y - x) dx

= [ yx - x²2 ]_x=0¹

f_Y(y) = y - 12

1 ≤ y ≤ 2

Question 5

Q5 (a): the decision

f_X(x) f_Y(y) = (32 - x)(y - 12)

≠ f(x, y)

Question 5

Q5 (b): E(X)

E(X) = ∫10 x (32 - x) dx

= [ 32 · x²2 - x³3 ]₀¹

= 34 - 13 = 512

Question 5

Q5 (b): E(X²)

E(X²) = ∫10 x² (32 - x) dx

= [ 32 · x³3 - x⁴4 ]₀¹

= 36 - 14 = 14

Question 5

Q5 (b): Var(X)

Var(X) = E(X²) - [E(X)]²

= 14 - (512)²

= 36 - 25144 = 11144

Question 5

Q5 (b): E(Y)

E(Y) = ∫21 y (y - 12) dy

= [ y³3 - 12 · y²2 ]₁²

= 73 - 34 = 1912

Question 5

Q5 (b): E(Y²)

E(Y²) = ∫21 y² (y - 12) dy

= [ y⁴4 - 12 · y³3 ]₁²

= 154 - 76 = 3112

Question 5

Q5 (b): Var(Y)

Var(Y) = E(Y²) - [E(Y)]²

= 3112 - (1912)²

= 372 - 361144 = 11144

Question 5

Q5 (b): E(XY), the integral

E(XY) = ∫10 ∫21 xy f(x, y) dy dx

= ∫10 ∫21 xy (y - x) dy dx

= ∫10 [ x y³3 - x² y²2 ]_y=1² dx

Question 5

Q5 (b): E(XY), the value

= ∫1x=0 (7x3 - 3x²2) dx

= [ 7x²6 - 3x³6 ]₀¹

E(XY) = 76 - 12 = 46 = 23

A slip on the exercise sheet

Question 5

Q5 (b): Cov(X, Y)

Cov(X, Y) = E(XY) - E(X) E(Y)

= 23 - (512 · 1912)

= 96 - 95144 = 1144

Question 5

Q5 (b): Cor(X, Y)

Cor(X, Y) = Cov(X, Y)σ₁ σ₂

= Cov(X, Y)√(Var(X) Var(Y))

= 1 / 144√((11 / 144) (11 / 144))

= 1 / 14411 / 144

Cor(X, Y) = 111

Part 7

Correlation from recorded data

Q6: lead time and demand

Question 6

Q6: lead time and demand

X1: average lead time, in months. X2: annual demand. 10 years.

X1	X2	X1	X2
6.5	103	6.9	104
4.3	83	5.8	106
6.9	116	7.3	109
6	97	4.5	92
6.9	112	6.3	96

Question 6

Q6: the two means

μ = ∑ni=1 x_in

μ₁ = 6.5 + 4.3 + ... + 6.310

= 61.410 = 6.14

μ₂ = 103 + 83 + ... + 9610

= 101810 = 101.8

Question 6

Q6: the lead time variance

S² = ∑ni=1 x_i² - n μ²n - 1

S₁² = 386.44 - 10 (6.14)²9

= 386.44 - 376.9969

= 1.05

σ₁ = √(1.05) = 1.025

Question 6

Q6: the demand variance

S₂² = 104520 - 10 (101.8)²9

= 104520 - 103632.49

= 98.6

σ₂ = √(98.6) = 9.93

Question 6

Q6: the covariance

Cov = ∑nj=1 X_1j X_2j - n μ₁ μ₂n - 1

∑ X_1j X_2j = (103 × 6.5)

+ ... + (96 × 6.3) = 6328.5

n μ₁ μ₂ = 10 × 6.14 × 101.8

= 6250.52

Cov = 6328.5 - 6250.529

= 8.66

Question 6

Q6: the correlation coefficient

ρ = Cov(X₁, X₂)σ₁ σ₂

= 8.661.025 × 9.93 = 0.85

A slip on the exercise sheet

Part 8

Common mistakes

and practice

Common mistakes

The variance term is f m², not (f m)²
Keep the mean unrounded until the variance is done
One matching point does not prove independence
The Poisson denominator is x!
Degrees of freedom: k - s - 1, with k counted after combining
A statistic larger than the table value means H0 is rejected

Practice

Q2: compare p(x, y) with the product of the two distributions at (2, 0)
Q4: compute f(2) and E from the formula
Q4: repeat the decision at α = 0.01
Q5: integrate f(x, y) over its whole range
Q3: repeat with logarithms at two decimals

Key Takeaways

From a frequency table: the mean uses Σ f m, the variance uses Σ f m²
Independent only if the product equals the joint function at every point
Lognormal data: take natural logarithms first
Chi-square: E = n f(x), combine small intervals, compare at k - s - 1
Var(X) is E(X²) minus [E(X)]², and Cov is E(XY) minus E(X)E(Y)
Cor = Cov / (σ1 σ2): close to 1 means strongly positive

Open this lesson

Mahmoud AbasInput Modeling: Estimation, Testing and Correlation

Job

Questions

Estimate parameters from data

Q1, Q3

Test a candidate distribution

Check dependence

Q2, Q5

Measure correlation

Q5, Q6

Frequency f

f × m

f × m²

153

160

Frequency f

f × m

f × m²

200

252

245

320

243

Frequency f

f × m

f × m²

300

121

Total

100

364

2080

Point

p(x, y)

Product

Match

(0, 0)

0 / 150

3 / 150

(1, 2)

15 / 150

Yes

(2, 3)

25 / 150

28 / 150

ln x

18.8

2.9

1.6

27.9

3.3

22.9

3.1

6.1

1.8

3.1

1.1

37.4

3.6

8.3

2.1

Observed O

f(x)

Expected E

0.0263

2.6

0.0956

9.6

0.1739

17.4

0.2110

21.1

0.1920

19.2

Observed O

f(x)

Expected E

0.1398

14.0

0.0848

8.5

0.0441

4.4

0.0201

2.0

0.0081

0.8

Observed O

f(x)

Expected E

0.0030

0.3

0.0010

0.1

Interval x

0, 1

12 + 10 = 22

2.6 + 9.6 = 12.2

7 to 11

7.6

Interval x

(O - E)² / E

0, 1

12.2

7.87

17.4

0.15

21.1

0.80

19.2

4.41

14.0

2.57

Interval x

(O - E)² / E

8.5

0.26

7 to 11

7.6

11.63

χ² 0.025

χ² 0.05

χ² 0.10

9.35

7.81

6.25

11.14

9.49

7.78

12.8

11.1

9.2

14.4

12.6

10.6

16.0

14.1

12.0

6.5

103

6.9

104

4.3

5.8

106

6.9

116

7.3

109

4.5

6.9

112

6.3