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ignou MTE 11 solved assignment 2024

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Last Date of Submission of IGNOU MTE-011 (BSC) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Semester Wise
January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam).
July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).

Title Name	IGNOU MTE 11 Probability and Statistics Solved Assignment 2024
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSC
Course Name	Bachelor in Science
Subject Code	MTE 11
Subject Name	Probability and Statistics
Year	2024
Session	-
Language	English Medium
Assignment Code	MTE-011/Assignmentt-1//2024
Product Description	Assignment of BSC (Bachelor in Science) 2024. Latest MTE 011 2024 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MTE-011 (BSC) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam). Semester Wise January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam). July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Assignment Code	MTE 11/2024

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Questions Included in this Help Book

Ques 1.

Which of the following statements are True or False? Give reasons for your answer.
i) If $y=ax-b,$ then the correlation coefficient between x and y does not exist and if it exists it is equal to zero.
ii) For a normal distribution mean, median and standard deviation are all equal.
iii) If $x\geq y$ then $y-x$ assume only non-positive values and hence $E(x)\leq E(y).$
iv) If two unbiased dice are rolled, then the probability of their same score being 6 is $\frac{1}{6}.$

v) If the random variable x follows a normal distribution with known mean µ and unknown variance $\sigma ^2$ then $\frac{x-\mu }{\sigma }$ is a statistic but ) (x −µ is not.

Ques 2.

The first three moments of a distribution about the value 2 are 1, 16 and – 40 respectively. Examine the Skewness of the distribution.

Ques 3.

Find the maximum likelihood estimator for the parameter λ of the Poisson distribution on the basis of a sample of size n . Also, find its variance.

Ques 4.

If x has an exponential distribution with parameter θ . Find the density function of $log_e\,x.$

Ques 5.

If $x\sim n(\mu ,\sigma ^2).$ Find the moment generating function of x − c where c is constant.

Ques 6.

Let p be the probability that a coin will fall head in a single toss in order to test $H_0:p=\frac{1}{2}$ against $H_1:p=\frac{3}{4},$ The coin is tossed 3 times and $H_0$ is rejected if more than 2 heads are obtained. Find the probability of type I and type II errors. Also obtain the power of the test.

Ques 7.

An unbiased die is rolled twice. Let $A_1$ denote the event: odd face roll on the first die, $A_2$ denote the event that total score is Odd. Check the independence of $A_1$ and . $A_2$

Ques 8.

Let $f(x,y)=x+y;0<x<1,0<y<1.$ Find (i) the correlation coefficient between x and y, and (ii) E( y / x) .

Ques 9.

Show that variance can be expressed in terms of the mutual differences $x_i-x_j$ of the observations i.e.

$S^2=\frac{1}{2n^2}\sum_{i=1}^{n}\sum_{j=1}^{n}(x_i-x_j)^2.$

Ques 10.

The joint probability distribution of x and y is given below:

$x\rightarrow$	$y\downarrow$	0	1	2
0		$\frac{1}{12}$	$\frac{1}{6}$	$\frac{1}{24}$
1		$\frac{1}{4}$	$\frac{1}{4}$	$\frac{1}{40}$
2		$\frac{1}{8}$	$\frac{1}{20}$	__
3		$\frac{1}{120}$	__	$\frac{1}{120}$

Find (i) ) $P(x=1,=2)$
(ii) ) $P(x=0,1\leq y<3)$
(iii) ) $P(x+y\leq1)$
(iv) ) $P(x>y)$

Ques 11.

Every clinical thermometer is classified into one of the four catergories A, B, C and D on the basis of inspection and test. From the past experience it is known that thermometers produced by a certain manufacturer are distributed among the four categories in the following proportions:

Category:	A	B	C	D
Proportion:	0.87	0.009	0.003	0.01

A new lot of 1336 thermometers is submitted by the manufacture for inspection and test and the following distribution of categories obtained:

Category:	A	B	C	D
No. of thermometers Reported:	1188	91	47	10

At 5% level of significance test whether this new lot of thermometers differ from the previous experience.

Ques 12.

The following information about advertising expenditures and sales of a company is given below:

	Advertising expenditure (x) Lakhs ( )	Sales (y) Lakhs ()
Average	10	90
Variance	9	144
Correlation Coefficient between x and y is 0.8.

What should be the advertising budget if the company wants to attain sales target of 120 Lakhs?

Ques 13.

The police plans to enforce speed limits by using radar traps of 4 different locations within the city. The radar traps at each of the locations $L_1,L_2,L_3$ and $L_4$ are operated 40 % 30% 20% and 30%, of time. If a person who is speeding on h, is way to work has probabilities of 0.2, 0.1, 0.5 and 0.2 respectively of passing through these locations. What is the probability that the person will receive a speeding challan.

Ques 14.

a) Find the mean and standard deviation for the following data:

Class interval	Frequency
0 – 10	5
10 – 20	10
20 – 30	14
30 – 40	15
40 – 50	6

b) State Chebychev’s inequality. Hence obtain the lower bound for $P[-1<x<9]$ if the E(x) and $E(x^2)$ of x are 4 and 20 respectively.

Ques 15.

Let X be a single observation from the p.d.f. $f(x,\theta )=\theta e^{\theta x},0\leq x<\infty .$

If X ≥1 is the critical region for testing $H_0:\theta=2$ against the alternative hypothesis $H_1:\theta=1,$ obtain the values of type I and type II errors.

Ques 16.

The probability that a student passes a Physics test is $\frac{2}{3}$
and the probability that the student passes both a Physics test and an English test is $\frac{14}{15}$ The probability that the
student passes at least one test is $\frac{4}{5}$ What is the probability that the student passes the English test?

Ques 17.

Draw the cumulative frequency curves for the following distribution:

Marks	No. of Students
0 – 10	4
10 – 20	8
20 – 30	11
30 – 40	15
40 – 50	12
50 – 60	6
60 – 70	3

From the graph, obtain the median.

Ques 18.

X , X and X3 is a random sample of size 3 from a population with mean µ and variance $\sigma ^2.$ $T_1,T_2$ . and $T_3$ are the estimators to estimate µ, and are given by ; $T1=X_1+X_2-X_3;T_2=2X_1+3X_2-4X_2$ and $T_3(\lambda X_1+X_2+X_3).$

i) Are T1 and T2 unbiased? Give reason.
ii) Find the value of λ such that T3 is unbiased.
iii) Which is the best estimator? State giving reasons.

Ques 19.

A group of 250 items with mean 15.6 and standard deviation $\sqrt{13.44}$ . has been divided into two groups. The first has 100 items with mean 15 and standard deviation 3. Find the standard deviation of the second group.

Ques 20.

A single observation was taken from a population with p.d.f. $f(x,\theta)=\frac{2}{\theta^2}(\theta-x),$ $0\leq x\leq\theta.$

Obtain $100(1-\alpha )%$ confidence interval for θ.

Ques 21.

For 10 observations on price $(X)$ and supply $(Y)$ the following data were obtained (in appropriate units): $\sum X=130,\sum Y=200,\sum X^2=2288,\sum Y^2=5506 \,$ $\sum XY=3467.$