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BMTC 101: Introduction to Probability and Statistics

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Title Name	IGNOU BMTC 101 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSCFMT
Course Name	Bachelor of Science (Mathematics)
Subject Code	BMTC 101
Subject Name	Introduction to Probability and Statistics
Year	2026
Session	-
Language	English Medium
Assignment Code	BMTC 101/Assignment-1/2026
Product Description	Assignment of BSCFMT (Bachelor of Science (Mathematics)) 2026. Latest BMTC 101 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BEGC-131 (BAG) 2025-26 Assignment is for January 2026 Session: 30th September, 2026 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
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BMTC 101 2026 - English

Assignment
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Course Code: BMTC-101
Assignment Code: BMTC-101/TMA/2026
Maximum Marks: 100

1. Which of the following statements are True or False? Give short proof or counter example in your answer.

i) If the correlation coefficient between X and Y is -0.8, then the correlation coefficient between 2X - 1 and -3Y - 1 is -0.48.

ii) If X and Y are independent binomial variates with parameters (n₁, p₁) and (n₂, p₂) respectively, then X + Y has binomial distribution with parameters (n₁ + n₂, p₁ + p₂).

iii) The function defined as
$$f(x) = \begin{cases} |x| ; & -1 < x < 1 \\ 0 ; & \text{otherwise} \end{cases}$
is a probability density function.

iv) For a normal distribution with mean $\mu$ and variance $\sigma^2$ , the hypotheses
$H_1 : \mu = \mu_0, \sigma^2 = 1$ and
$H_2 : \mu = \mu_0, \sigma^2 \ge 1$ are simple hypotheses.

v) In a problem of testing of a simple hypothesis against a simple alternative, if the probability of type-I error is known to be 0.06, then the power of the test will be 0.94.

2. The mean I.Q. of a large number of children of age 14 was 100 and standard deviation 16. Assuming that the distribution was normal, find

$\quad$ i) the percentage of children having I.Q. under 80.

ii) the limits in which the I.Q. of the middle 40% of the children will lie.

$\quad$ You may like to use the following values:

$\quad P(Z > 1.25) = 0.1056$

$\quad P(Z < -0.525) = 0.3$

3. 6 observations on (X, Y) yielded the following data:

$$\sum X_i = 30, \sum Y_i = 180, \sum X_i Y_i = 1000,$

$$\sum X_i^2 = 200, \sum Y_i^2 = 5642.$

$\quad$ i) Determine the correlation coefficient between X and Y.

ii) Given $X = 10$ , what will be the predicted value of Y?

iii) Given $Y = 15$ , what will be the predicted value of X?

4. A die is thrown 60 times with the following results:

Face of die	1	2	3	4	5	6
Frequency	8	7	12	8	14	11

Test that the die is unbiased at $5\%$ level of significance. Given that at 5, 6 and 7 d.f. the value of $\chi^2$ are 11.070, 15.592 and 14.067 respectively.

5. Consider the joint probability density function

$$f(x, y) = y^2 e^{-y(x+1)}; x \geq 0, y \geq 0.$

Are both x and y regressions linear? Give reasons for your answer.

6. a) The mean and standard deviation of a variable x are m and $\sigma$ respectively. Obtain the mean and standard deviation of $\frac{(ax+b)}{c}$ , where a, b and c are constants.

$\quad$ b) If X is a random variable such that $E(X) = 3$ and $E(X^2) = 13$ , determine a lower bound for P(-2 < X < 8).

7. a) Let E₁, E₂, E₃ and E₄ be arbitrary events. Write the following events in set notations:

$\quad$ i) not more than one of E₁, E₂, E₃, E₄.
$\quad$ ii) one and only one of E₁, E₂, E₃, E₄.
$\quad$ iii) E₁ and at least one of E₂, E₃, E₄.
$\quad$ iv) none of E₂, E₃ and E₄ using E₁.

$\quad$ b) Let the probability density function of r.v. X be
$$f(x) = \begin{cases} 1 + x & ; \ -1 < x \le 0 \\ 1 - x & ; \ 0 < x < 1 \\ 0 & ; \ \text{otherwise} \end{cases}$
$\quad$ and if $u = X$ and $v = X^2$ , find Cov(u, v). Also check the independence of u and v.

8. a) For a mesokurtic distribution with standard deviation 5, find fourth central moment m₄.

$\quad$ b) The probability that a card will have a flat tyre while crossing a certain bridge is 0.00005. Find the probability that, among 10,000 cars crossing the bridge,

$\quad$ i) exactly two cars will have a flat tyre.
$\quad$ ii) at most two cards will have a flat tyre.

9. a) Let X₁ be an observation from an exponential distribution with the p.d.f.

$$f(x) = \frac{1}{\theta} e^{-x/\theta}; x > 0$

Test the null hypothesis that the mean of the distribution is $\theta = 2$ against the alternative hypothesis that is $\theta = 5$ . The null hypothesis is accepted if and only if the observed value of the random variable is less than 3. Find the probabilities of type-I and type-II errors.

b) The mean and standard deviation of 20 items is found to be 10 and 2 respectively. At the time of checking it was found that one item having value 8 was incorrect. Calculate the mean and standard deviation if the wrong item is omitted.

10. a) Let X be a gamma variable with parameters αand λ, having E(X ) = 6 and Var(X ) = .3 Find αand λ. Also, find the m.g.f. of a gamma variable, and hence verify that mean of X is 6 and variance of X is 3 using m.g.f.

b) For married couples living in a certain locality, the probability that the husband will vote in a school board election is 0.21, the probability that they both will vote is 0.15. What is the probability that

i) at least one of them will vote?

ii) neither of them will vote?

BMTC 101 2026 - Hindi

सत्रीय कार्य

पाठ्यक्रम कोड: BMTC-101

सत्रीय कार्य कोड : BMTC-101/TMA/2026

अधिकतम अंक: 100

1. निम्नलिखित में से कौन-से कथन सत्य हैं और कौन-से असत्य? अपने उत्तर में संक्षिप्त उपपत्ति या प्रत्युदाहरण दीजिए। (10)

i) यदि X और Y के बीच सहसंबंध गुणांक -0.8 है, तो 2X - 1 और -3Y - 1 के बीच सहसंबंध गुणांक -0.48 होगा।

ii) यदि X और Y क्रमशः (n₁, p₁) और (n₂, p₂) प्राचलों वाले स्वतंत्र द्विपद चर हैं, तो X + Y प्राचल (n₁ + n₂, p₁ + p₂) वाला द्विपद बंटन होगा।

iii) $f(x) = \begin{cases} |x|, & -1 < x < 1 \\ 0, & \text{otherwise} \end{cases}$ द्वारा परिभाषित फलन एक प्रायिकता घनत्व फलन है।

iv) माध्य $\mu$ और प्रसरण $\sigma^2$ , वाले एक प्रसामान्य बंटन के लिए, परिकल्पनाएँ
$H_1 : \mu = \mu_0, \sigma^2 = 1$ और
$H_2 : \mu = \mu_0, \sigma^2 \ge 1$ सरल परिकल्पनाएँ हैं।

v) सरल वैकल्पिक के विरुद्ध सरल परिकल्पना के परीक्षण की समस्या में, यदि टाइप-I अशुद्धि की प्रायिकता 0.06 है, तो परीक्षण की क्षमता 0.94 होगी।

2. 14 वर्ष की आयु के बच्चों की एक बड़ी संख्या का माध्य I.Q. 100 और मानक विचलन 16 था। मान लीजिए कि बंटन प्रसामान्य था, तो निम्नलिखित ज्ञात कीजिए :

i) 80 से कम I.Q. वाले बच्चों का प्रतिशत।

ii) वह सीमा जिसके लिए बीच में 40% बच्चों की I.Q. होगी।

आप निम्नलिखित मानों का प्रयोग कर सकते हैं :

$P(Z > 1.25) = 0.1056$

$P(Z < -0.525) = 0.3$

3. (X, Y) पर किए गए 6 प्रेक्षणों से प्राप्त आँकड़े निम्नलिखित हैं :

$\sum X_i = 30, \sum Y_i = 180, \sum X_i Y_i = 1000,$

$\sum X_i^2 = 200, \sum Y_i^2 = 5642.$

i) X और Y के बीच सहसंबंध-गुणांक ज्ञात कीजिए।

ii) दिया गया है $X = 10$ , तब Y का प्रागुक्त मान क्या होगा?

iii) दिया गया है $Y = 15$ , तो X प्रागुक्त मान क्या होगा?

4. एक पासे को 60 बार फेंकने से निम्नलिखित परिणाम प्राप्त होते हैं :

पासे का मुख	1	2	3	4	5	6
बारम्बारता	8	7	12	8	14	11

5% सार्थकता स्तर पर परीक्षण कीजिए कि पासा अनभिनत है। दिया गया है कि 5, 6 और 7 स्वातंत्र्य कोटि के लिए $\chi^2$ के मान क्रमशः 11.070, 15.592 और 14.067 हैं।

5. निम्नलिखित संयुक्त प्रायिकता घनत्व फलन लीजिए :

$$f(x, y) = y^2 e^{-y(x+1)}; x \ge 0, y \ge 0.$

क्या दोनों x और y समाश्रयण रैखिक हैं? अपने उत्तर का कारण दीजिए।
6. a) एक चर x का माध्य और मानक विचलन क्रमशः m और $\sigma$ है। $\frac{(ax + b)}{c}$ , का माध्य और मानक विचलन ज्ञात कीजिए जबकि a, b और c अचर हैं।

b) यदि एक यादृच्छिक चर X इस प्रकार है कि $E(X) = 3$ और $E(X^2) = 13$ है, तो P(-2 < X < 8) का निम्न परिबंध ज्ञात कीजिए।

7. a) मान लीजिए कि E₁, E₂, E₃ और E₄ स्वैच्छिक घटनाएँ हैं। निम्नलिखित घटनाओं को समुच्चय संकेतनों में लिखिए:

i) E₁, E₂, E₃ और E₄ में एक से अधिक नहीं।

ii) E₁, E₂, E₃ और E₄ में से केवल एक।

iii) E₁ और कम-से-कम E₂, E₃, E₄ में से एक।

iv) E₁ के साथ E₂, E₃ और E₄ में से कोई नहीं।

b) मान लीजिए कि r.v. X का प्रायिकता घनत्व फलन निम्नलिखित है :

$f(x) = \begin{cases} 1 + x, & -1 < x \le 0 \\ 1 - x, & 0 < x < 1 \\ 0, & \text{otherwise} \end{cases}$
और यदि $u = X$ और $v = X^2$ हो, तो Cov(u, v) ज्ञात कीजिए। u और v के स्वातंत्र्य की भी जाँच कीजिए।

8. a) मानक विचलन 5 वाले मध्यककुदी बंटन के लिए चतुर्थ केंद्रीय आघूर्ण m₄ ज्ञात कीजिए।

b) किसी पुल को पार करते समय एक कार के टायर सपाट होने की प्रायिकता 0.00005 है। पुल पर करने वाली 10,000 कारों के लिए वह प्रायिकता ज्ञात कीजिए जबकि

i) ठीक 2 कारों के टायर सपाट होंगे।

ii) ज्यादा से ज्यादा 2 कारों के टायर सपाट होंगे।

9. a) मान लीजिए कि X₁ एक चरघातांकी बंटन का एक प्रेक्षण है, जिसका प्रायिकता घनत्व फलन

$$f(x) = \frac{1}{\theta} e^{-x/\theta}; x > 0$

है। निराकरणीय परिकल्पना कि बंटन का माध्य $\theta = 2$ है की प्रतिकूल वैकल्पिक परिकल्पना कि माध्य $\theta = 5$ है, का परीक्षण कीजिए। निराकरणीय परिकल्पना केवल तभी स्वीकार की जाती है यदि और केवल यदि यादृच्छिक चर का प्रेक्षित मान 3 से कम हो। टाइप-I और टाइप-II त्रुटियों की प्रायिकताएँ ज्ञात कीजिए।

b) 20 पदों का माध्य और मानक विचलन क्रमशः 10 और 2 पाया गया है। जाँच करते समय यह पाया गया कि मान 8 वाला एक पद गलत है। यदि गलत पद को हटा दिया जाए तो माध्य और मानक विचलन परिकलित कीजिए।

10. a) मान लीजिए X प्राचल $\alpha$ और $\lambda$ वाला एक गामा चर है जिसके $E(X) = 6$ और $Var(X) = 3$ हैं। $\alpha$ और $\lambda$ ज्ञात कीजिए। गामा चर का आघूर्ण जनक फलन भी ज्ञात कीजिए और इस प्रकार सिद्ध कीजिए कि आघूर्ण जनक फलन से X का माध्य 6 और X का प्रसरण 3 है।

b) एक मुहल्ले में रहने वाले वैवाहिक जोड़ों में, पति के एक स्कूल बोर्ड चुनाव में वोट देने की प्रायिकता 0.21 है, पत्नी के चुनाव में वोट देने की प्रायिकता 0.28 है और दोनों के वोट देने की प्रायिकता 0.15 है। इस बात की प्रायिकता क्या होगी कि

i) उनमें से कम से कम एक वोट देगा?

ii) उनमें से कोई भी वोट नहीं देगा?

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