Elements of Probability B.Sc Question Paper : mlsu.ac.in
University : Mohan Lal Sukhadia University
Degree : B.Sc
Department : Statistics
Subject : Elements of Probability
Year : I
Document Type : Question Paper
Website : mlsu.ac.in
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Elements of Probability Question Paper :
First Year Examination of the Three Year Degree Course, 2001
(Common for the Faculties of Arts & Science)
Statistics Paper II(Elements of Probability)
Related : Mohan Lal Sukhadia University Official Statistics & Numerical Methods B.Sc Question Paper : www.pdfquestion.in/5826.html
Time – Three Hours Maximum Marks – 50
Attempt Five question in all, selecting ONE question from each unit. All questions carry equal marks.
SECTION A
1. (a) Define probability and explain the importance of this concept in Statistics. (b) What is the probability of having a knave and a queen when two cards are drawn from a pack of 52-
2. (a) State and prove the theorem of multiplication of probability.
(b) A card is drawn from a well shuffled pack of playing cards. What is the probability that it is either a spade or an ace-
3. (a) State and prove Baye’s theorem.
(b) In a bolt factory machines. A, B and C manufacture respectively 25% , 35% and 40% of the total. Of their output 5, 4 and 2 percents are defective bolts. A bolt is drawn at random from the product and What is the probability that it was manufactured by machine A-
SECTION B
4. (a) Explain random variable, discrete random variable, continuous random variable and their distribution functions.
(b) From a lot of 10 items containing 3 defectives, a sample of 4 items is drawn at rndom. Let the random variable X denote the number of defective items in the sample. Answer the following :
(i) Find the probability distribution of X.
(ii) Find P (X < 1), P (X < 1) and P (0 < x< 2).
5. In a continuous distribution whose relative frequency density is given by :
F(x) = y0 x (2 – x); 0 < x <2. Find mean , variance, b1 and b2 and hence, show that the distribution is symmetrical.
6. What do you understand by marginal and conditional distribution- For the following bivariate probability distribution of X and Y find :
(i) P ( X < 1, Y = 2)(ii) P ( X < 1)
(iii) P ( Y = 3)(iv) P ( Y < 3) and(v) P (X < 3, Y < 4).
SECTION C
7. (a) What is Sheppard’s Correction- When is it used- Find the corrected moments of the following values if the magnitude of the class interval is m 2 = 43.35, m3 = 9.77 and m4 = 5508.56
(b) Define Covariance. Show that covariance of two independent random variable is always zero. Is the converse true- Give an example to support your answer.
8. (a) Define mathematical expectation of a random variable. Show that the expectation of product of two independent variable is the product of their expectation. Is the condition of independence necessary? If not, what is the necessary condition?
(b) X and Y are independent variables with means 10 and 20 and variances 2 and 3 respectively. Find the variance of 3X + 4Y.
9. (a) Define moment generating function of a distribution. Explain how it helps to find moments of the distribution.
(b) Obtain the moment generating function of the following : P(x) = 1/2x; x = 1, 2, 3 … … . Hence find mean and variance.
10. (a) Define Cumulant Generating function. State and prove additive property of cumulants.
(b) Define the characteristic function of a random variable. Write down its properties.
Syllabus :
Paper -II : Probability Theory
TIME : 3 hours
UNIT – I :
Random experiment, sample space, events, elements of an event, union and intersection of events, mutually exclusive, exhaustive, independent and equally likely events.
Classical and Statistical definitions of probability and simple problems, Axiomatic approach to probability. Addition law of probability for two or more events.
UNIT – II :
Conditional probability, Multiplication law of probability, Statistical independence of events, Baye’s theorem and its simple applications.
UNIT – III :
Random Variable Discrete and continuous random variables, Probability mass and density functions,- joint, marginal and conditional probability functions, Distribution functions.
UNIT -IV :
Mathematical Expectation Definition of expectation, Addition and Multiplication laws of expectation, Moments in terms of expectation, variance and covariance for the linear combination of random variables. Elementary idea of conditional expectation. Schwartz’s inequality.
UNIT – V :
Moment generating and Cumulants generating functions with properties, Characteristic function with properties (without proof).