मानचित्र मापनी के विभिन्न प्रकारों और निर्माण विधि को उपयुक्त उदाहरणों सहित समझाइए।
See Answer →Evaluate the integral using beta and gamma functions.
Evaluate the integral , where
, by considering D as a region of Type I and then as a region of Type II.
If be a function defined by
,
. Show that f is Riemann integrable using both definitions. Also, verify that the results of both definition match.
Find the equation of a line passing through points A(0, 1, 1) and B(1, 1, 0). Also, find the coordinates of a point on this line which is at a distance of 10 units from point A opposite to the side of point B.
(2 + 3 + 5)
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In R we have a built-in data set "trees". A screenshot of the last four rows together with the R code to obtain it is given as follows. To get more detail about this data set you can run ?trees command on R console.
```
> tail(trees, 4)
Girth Height Volume
28 17.9 80 58.3
29 18.0 80 51.5
30 18.0 80 51.0
31 20.6 87 77.0
```
Note that all the three variables of this data set are numeric. So, assuming each row of this data set is a point in 3-dimension. Find the distances between the points corresponding to the and the
rows using the Manhattan and Chebyshev distance formula.
If f is a function from set X to a set Y then is it possible ,
? where
and
If and
be two independent gamma distributions and
and
then find the distribution of V.
Explain the procedure of assigning probability in the discrete world of probability theory.
See Answer →In an election there are two candidates. Being a statistician, you are interested in predicting the result of the election. So, you plan to conduct a survey. Using the learning skill of this course answer the following question. How many people should be surveyed to be at least 95% sure that the estimate is within 0.05 of the true value?
See Answer →Solve the problem in part (a) simply by replacing 10 minutes by 5 minutes all other things are the same.
See Answer →a) Suppose two friends Anjali and Prabhat trying to meet for a date to have lunch say between 1 pm to 2 pm. Suppose they follow the following rules for this meeting:
Each of them will arrive either on time or 10 minutes late or 20 minutes late or 30 minutes late or 40 minutes late or 50 minutes late or 1 hour late. All these arrival times are equally likely for both of them.
Whoever of them reaches first will wait for the other to meet only for 10 minutes. If within 10 minutes the other does not reach, he/she leaves the place and they will not meet.
Find the probability of their meeting.
Differentiate between the autoregressive and moving average models of time series.
See Answer →The marketing manager of a company recorded the number of mobiles sold quarterly for which are given in the following table:
I’ve transcribed the table from your image exactly as it appears.
Quarterly Data Table
| Year / Quarter | $Q_1$ | $Q_2$ | $Q_3$ | $Q_4$ |
|---|---|---|---|---|
| 2018 | 48 | 41 | 60 | 65 |
| 2019 | 58 | 52 | 68 | 74 |
| 2020 | 60 | 56 | 75 | 78 |
(i) Find the quarterly seasonal indexes for the mobile sold using the ratio to trend method.
(ii) Do seasonal forces significantly influence the sale of mobile? Comment.
(iii) Also find the deseasonalised values.
See Answer →The marketing manager of a company recorded the number of mobiles sold quarterly for which are given in the following table:
I’ve transcribed the table from your image exactly as it appears.
Quarterly Data Table
| Year / Quarter | $Q_1$ | $Q_2$ | $Q_3$ | $Q_4$ |
|---|---|---|---|---|
| 2018 | 48 | 41 | 60 | 65 |
| 2019 | 58 | 52 | 68 | 74 |
| 2020 | 60 | 56 | 75 | 78 |
(i) Find the quarterly seasonal indexes for the mobile sold using the ratio to trend method.
(ii) Do seasonal forces significantly influence the sale of mobile? Comment.
(iii) Also find the deseasonalised values.
See Answer →Consider the time series model
y₁=5+0.6y1-0.32 +8
where &, ~N[0,1]
(i) State whether the process is stationary, giving reasons.
(ii) Find the mean of the process.
(iii) Obtain the variance of the process.
(iv) Derive the autocorrelation function up to lag 2 and sketch the correlogram.
See Answer →In a manufacturing process, 4 items are inspected every hour for 10 consecutive hours.
The measured quality characteristic (in mm) is given below:
The measured quality characteristic (in mm) is given below.
| Hour | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 50 | 52 | 51 | 49 |
| 2 | 51 | 50 | 53 | 52 |
| 3 | 49 | 50 | 48 | 51 |
| 4 | 52 | 54 | 53 | 51 |
| 5 | 50 | 49 | 51 | 50 |
| 6 | 53 | 52 | 54 | 55 |
|---|---|---|---|---|
| 7 | 48 | 49 | 50 | 47 |
| 8 | 51 | 52 | 50 | 51 |
| 9 | 54 | 55 | 53 | 54 |
| 10 | 50 | 51 | 49 | 50 |
(i) Construct conreol chart for variability and mean and comment on the state of statistical control. If the process is out of control, obtain revised control limits.
(ii) Given specification limits 50 ± 3, compute the process capability index (Cpk) and interpret the result.
See Answer →In a manufacturing process, 4 items are inspected every hour for 10 consecutive hours.
The measured quality characteristic (in mm) is given below:
The measured quality characteristic (in mm) is given below.
| Hour | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 50 | 52 | 51 | 49 |
| 2 | 51 | 50 | 53 | 52 |
| 3 | 49 | 50 | 48 | 51 |
| 4 | 52 | 54 | 53 | 51 |
| 5 | 50 | 49 | 51 | 50 |
See Answer →
A system consists of six independent components arranged as follows:
Two components with reliabilities 0.9 and 0.8 connected in series.
This series combination is connected in parallel with a component of reliability 0.7.
The resulting subsystem is connected in series with two components of reliabilities 0.85 and 0.95.
Draw the reliability block diagram and calculate the overall reliability of the system.
See Answer →A factory purchases bolts in lots of 800. Acceptance is decided using a single-sampling plan with sample size n = 20 and acceptance number c = 3. Assume that 2% defective items are considered acceptable quality and 7% defective items are considered unacceptable quality.
Find:
(i) The probability of accepting a lot when the incoming quality level is 5% defective.
(ii) The Average Outgoing Quality (AOQ), assuming rejected lots are completely screened and defectives are replaced.
(iii) The Average Total Inspection (ATI).
See Answer →