Homework #12
EE 503: Summer 2026
Assigned: 03 August
Due: Monday, 10 August at 12:00
BrightSpace Assignment: Homework 12
Daily Derivation
Derive the system population mean \(E[N]\) and \(V[N]\) from the Global Balance Equations of the \(M/M/1\) queue.
Prove that the Ordinary Least Squares estimator \(\hat{\beta}^{\text{OLS}}\) is linear in \(Y\) if \(Y = X \beta + \epsilon\) and \(\epsilon \sim N(\mathbf{0}, \sigma^2 \mathbf{I})\): \(\hat{\beta}^{\text{OLS}} = (X^T X)^{-1} X^T Y\). Show that \(\hat{\beta}^{\text{OLS}} \sim N(\beta, \sigma^2 (X^T X)^{-1})\).
Write your solutions to these homework problems. Submit Handout and Book separately on BrightSpace by the due date. Show all work and box answers where appropriate. Do not guess.
Handout
Problem 1
Use a statistical software package (e.g. SPSS or Python) to compute the sample statistics \(\overline{x}\), \(\overline{y}\), \(s_x\), \(s_y\), \(s_x^2\), \(s_y^2\), \(s_{xy}\), \(r_{xy}\), \(r_{xy}^2\) from the following real data on the “sweetness” \(Y\) and the pectin content \(X\) of pressed orange juice from 24 production runs:
| Run | Sweetness index | Pectin (parts per million) |
|---|---|---|
| 1 | 5.20 | 220.00 |
| 2 | 5.50 | 227.00 |
| 3 | 6.00 | 259.00 |
| 4 | 5.90 | 210.00 |
| 5 | 5.80 | 224.00 |
| 6 | 6.00 | 215.00 |
| 7 | 5.80 | 231.00 |
| 8 | 5.60 | 268.00 |
| 9 | 5.60 | 239.00 |
| 10 | 5.90 | 212.00 |
| 11 | 5.40 | 410.00 |
| 12 | 5.60 | 256.00 |
| 13 | 5.80 | 306.00 |
| 14 | 5.50 | 259.00 |
| 15 | 5.30 | 284.00 |
| 16 | 5.30 | 383.00 |
| 17 | 5.70 | 271.00 |
| 18 | 5.50 | 264.00 |
| 19 | 5.70 | 227.00 |
| 20 | 5.30 | 263.00 |
| 21 | 5.90 | 232.00 |
| 22 | 5.80 | 220.00 |
| 23 | 5.80 | 246.00 |
| 24 | 5.90 | 241.00 |
What is the linear regression equation of the sweetness \(Y\) as a function of the pectin content \(X\)? Plot a “scatter diagram” with the sweetness index on the vertical axis and the pectin content of the juice on the horizontal axis. Turn in all printouts as appendices.
Problem 2
Repeat the above regression for 24 or more paired samples \((x_i, y_i)\) from some online database that you find and duly cite.
Book
Leon-Garcia
Chapter 6: 6.71-6.72