Seven successive observations on a stationary time-series are as follows:
12, 14, 13, 10, 15, 12, 15
(a) Calculate auto-covariances C0, C1, C2, C3 and C4.
(b) Calculate auto-correlation coefficients r1, r2, r3 and r4.
(c) Plot the correlogram.
(a) Auto-covariances C0,C1,C2,C3,C4C_0, C_1, C_2, C_3, C_4
To calculate the auto-covariances, we need the mean of the series first.
Given observations: 12, 14, 13, 10, 15, 12, 15
Step 1: Calculate the mean of the series:
Xˉ=12+14+13+10+15+12+157=917=13\bar{X} = \frac{12 + 14 + 13 + 10 + 15 + 12 + 15}{7} = \frac{91}{7} = 13
Now, we can calculate the auto-covariances. The general formula for CkC_k (auto-covariance) is:
Ck=1n−k∑t=1n−k(Xt−Xˉ)(Xt+k−Xˉ)C_k = \frac{1}{n-k} \sum_{t=1}^{n-k} (X_t - \bar{X})(X_{t+k} - \bar{X})
For C0C_0 (auto-covariance at lag 0):
C0=17∑t=17(Xt−Xˉ)2C_0 = \frac{1}{7} \sum_{t=1}^{7} (X_t - \bar{X})^2 C0=17[(12−13)2+(14−13)2+(13−13)2+(10−13)2+(15−13)2+(12−13)2+(15−13)2]C_0 = \frac{1}{7} \left[ (12 - 13)^2 + (14 - 13)^2 + (13 - 13)^2 + (10 - 13)^2 + (15 - 13)^2 + (12 - 13)^2 + (15 - 13)^2 \right] C0=17[1+1+0+9+4+1+4]=207≈2.857C_0 = \frac{1}{7} \left[ 1 + 1 + 0 + 9 + 4 + 1 + 4 \right] = \frac{20}{7} \approx 2.857
For C1C_1 (auto-covariance ________ ________ __________ __________ ___.
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Seven successive observations on a stationary time-series are as follows:
12, 14, 13, 10, 15, 12, 15
(a) Calculate auto-covariances C0, C1, C2, C3 and C4.
(b) Calculate auto-correlation coefficients r1, r2, r3 and r4.
(c) Plot the correlogram.
In a railway marshalling yard, goods trains arrive at a rate of 36 trains per day. Assuming that the inter-arrival and service time distributions both follow exponential distribution with an average of 30 minutes, calculate the following:
(i) Traffic intensity
(ii) The mean queue length
(iii) Probability that the queue size exceeds
State whether the following statements are True or False. Give reasons in support of your answers.
(a) The solution of a transportation problem with 5 rows (supplies) and 4 columns (destinations) is feasible if number of possible allocations are 8.
(b) The moving averages of suitable period in a time-series are free from the influences of seasonal and cyclic variations.
(c) If the basic solutions for a system of equations are (-2, 0, 1), (0, 1, 3), (-2, 3, 0), then only (0, 1, 3) is feasible.
(d) In the stepwise selection method of multiple regression model, once a variable enters in the model then it always remains in the model.
(e) An enterprise requires 1000 units per month. The ordering cost is estimated to be 50 per order. The purchase price is 20 per unit and the carrying cost per unit is 10% of it. Then the economic lot size to be ordered is 775.
The following table represents the sales (in thousands) of mobile sets of a shop for 16 quarters over four years:
| Year | Quarter | |||
| Q1 | Q2 | Q3 | Q4 | |
| 2011 | 554 | 590 | 616 | 653 |
| 2012 | 472 | 501 | 521 | 552 |
| 2013 | 501 | 531 | 553 | 595 |
| 2014 | 403 | 448 | 460 | 480 |
A company has three production facilities S1, S2 and S3 with production capacity of 7, 9 and 18 units (in 100s) per week of a product, respectively. These units are to be shipped to four warehouses D1, D2, D3 and D4 with requirement of 5, 6, 7 and 14 units (in 100s) per week, respectively. The transportation costs (in Rs) per unit between factories to warehouses are given in the table below:
| Dl | D2 | D3 | D4 | Capacity | |
| S1 | 19 | 30 | 50 | 10 | 7 |
| S2 | 70 | 30 | 40 | 60 | 9 |
| S3 | 40 | 8 | 70 | 20 | 18 |
| Demand | 5 | 8 | 7 | 14 | 34 |
Use the penalty (Big M) method to solve the following LP problem:
Minimise Z 5x1+ 3x2
Subject to the constraints:
A researcher is interested in developing a linear model for the electricity consumption of a household having an AC (1.5 ton) so that she can predict the electricity consumption. For this purpose, she selects 25 houses and records the electricity consumption (in kWh), size of house (in square feet) and AC hours for one month during summers. The results obtained are:
SS(B0) = 12526.08, SS(B0,B1) = 17908.47, SS(B0,B2) = 17125.23, SS(B0, B1, B2) = 18079.0,
Build a regression model by selecting appropriate regressors in the model using the Stepwise Selection method.