1. Solve any four :
a) Slate different approaches to probability. State axioms of probability. 5
b) State and prove binomial theorem. 5
c) Suppose a license plate contain 3 letters followed by 4 digits. How many different license plates can be printed if:
a) Only the letters are repeated ?
b) Only the digits can be repeated ? 5
d) In how many ways 6 rings be worn on 4 fingers when
i) There can be only one ring on each finger.
ii) There can be any number of rings on each finger. 5
e) Solve the recurrence relation an - "-a^, + 2()an_2 = 0 to the initial conditions
a( = 2. 5
2. a) Find how many numbers between I and 500 both inclusive are divisible by
either 2 or 3 or 7 or 11. 7
b) Prove that
C(n, 1) + C(n, 3) + .... = C(n, 0) + C(n, 2) + ... = 2"'1. 8
3. a) Determine the discrete numeric function corresponding to generating function 8
I
A(z)=
5-6z + z2
b) hind the number of non negative integer solutions of the equation X, +X2+ X3 = 28
such that 4 < X, < 21, 5 <, ^ < 22, 6 £ X3 < 23. 7
4. a) Explain the following terms with examples. 6
i) Random Variable ii) Sample Space iii) Events
b) The following table represents the joint probability distribution of the discrete
random variable (X, Y). 8
1 2 3
1 k 5k 3k
2 2k 6k 7k
3 9k 10k Ilk
Find
i) k
ii) Conditional probability distribution of X given Y = 1
iii) Conditional probability distribution of Y given X = 2 and
iv) P(X + Y = 4)
c) Find mean and variance of Poisson distribution. 6
5. a) State Baye's theorem and solve the following. There arc 5 boys and 3 girls in
room No. 1 and there are 7 boys and 3 girls in room No. 2. A girl from one of
the two rooms laughed loudly. What is the probability the girl who laughed
loudly was from room No. 2 ? 7
b) A random variable X has following : N1 (t) = 2/(2 -1).
Find i) E(X) and ii) Var (X). 8
6. a) Find mean and variance of continuous uniform distribution. 7
b) The joint probability density function of (X, Y) is given by :
f(X,Y) =2e"xe"2y,0< x <oo,0<y <oo
■ 0 otherwise Compute :
i) P(X > I. Y < 1)
ii) P (X < Y) 8
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[3180]-205
MT-21: PROBABILITY AND COMBINATORICS (New) (2005 Pattern)
ime: 3 Hours Max. Marks: 70
N.B.: I) Question No. I and Question No. 4 are compulsory.
2) Solve any one from Question No. 2 and 3 and any one question from Question No. 5 and Question No. 6.
3) Figures to the right indicate full marks.
a) State and prove the Principle of Inclusion and Exclusion (PIE). 7
b) How many seven-place secret codes are possible when three of the entries are letters and four are digits. Repetition of digits and letters are allowed. 7
c) Solve the following Recurrence Relation 6
I «.-Vi + 2°v2 = 2 (5)n r —,—p fipa
a) If six people attended a party, where before joining the party they deposit
their hats in a check room. After the party the hats gel mixed up and the six
gentlemen picked their hats at random manner. What is the probability that
none of them receives their own hat 7
f2n" 2
+ 2
+ +
n 0 1 n
, J 1 J
b) Show the following by using combinatoric arguments: 8
(2n) (nf (n)2 (n^2
; n>r
ii)
i)
+ + fr + 2] + +
r r
V « r r
- r + 1
) Determine the discrete numeric function for which the generating function is 4-4z + z*i
) Find the number of positive integer solutions of the equation:
X! + x2 + x3 = 21; Xj > 2, x-, > 4, x3 > 5 ) Find the coefficient of x6 y8 z'° in the expansion of (2x3 - 3y2 - 5z)16.
4. a) State moment generating function and find moment generating function of I
Gamma distribution.
b) Show that Poisson distribution is a limiting case of Binomial distribution.
c) Solve the following problem using Baye's theorem.
Of the eggs supplied to a co-operative 30%, 20%, 35% and 15% come from the poultry farms A, B, C and D respectively. Rotten eggs account for 2%M 1%, 2.5%, and 1% of the supplies by A, B, C, D respectively. An egg \m taken at random and found to be defective. What is the probability that
i) It was supplied by A ii) It was supplied by C
5. a) State and prove mcmoryless/forgetfullness property of Exponential
Dislri bulion.
b) The joint probability function for the random variables X and Y is given below:
\ Y X \ 0 1 2
0 1/
/8 i/
/9 1/
/6
1 1/ /9 1/ /IK 1/
/v
2 1/ 1/
/6 i/
/I8
Find:
i) The marginal probability distribution of X and Y
ii) P [X> 1/ Y > 1]
iii) Are X and Y independent ?
6. a) The joint density function of (X, Y) is given by
f (x, y) = 2e~* e-2y ; 0 < x < oo
0 0 < y < oo
otherwise
compute:
i) P(X> 1, Y< 1) ii) P(X<Y)
b) Calculate mean and variance of Geometric distribution.
MT21 : PROBABILITY AND COMBINATORICS (New) (2005 Pattern)
Time : 3 Hours Max. Marks : 70
N.B.: i) Question No. 1 and question No. 4 are compulsory.
ii) Solve any one question from question nos. 2 and 3.
iii) Solve any one question from question nos. 5 and 6.
iv) Figures to the right indicate full marks.
1. a) Stale and prove Derangement theorem.
b) A shop sells six different flavours of ice-creams. In how many ways a customer
choose 4 ice-cream cones if :
i) they are not necessarily of different flavours.
ii) they contain only 3 different flavours.
c) Solve the recurrence relation a 5a , + 6a =5".
d) In how many ways we can arrange the alphabets of the word 'ARRANGE' so that
i) Two A's arc always together.
ii) Two A's are together and two R are not together. (5 Marks each)
2. a) Find number of non-negative integer solutions of the equation
x, + x2 + x3 = 17 if 2 < x, <8, 3 < <8f 4<x3<8. 7
b) How many ways are there to distribute eight balls into six boxes with the first two boxes collectively having atmost four balls if:
i) the balls are identical
ii) the balls are distinct. 8
3. a) Determine the discrete numeric function corresponding to generating function. 7
A(z)=-^T 4-4z + z2
b) Find coefficient of x4yV' in the expansion of (2x2 - y - 3z2)8. 4
c) If 4 Americans, 3 Frenchmen and 3 Englishmen are to be seated for dinner. How many ways they can sit on circular table if :
i) there is no restriction
ii) same nationality must sit next to each other ? 4
P.T.O.
4. a) Define the following events with illustration.
i) Exhaustive events ii) Mutually exclusive events
iii) Equally likely events iv) Independent events. 8
b) In a basket there are 12 mangoes of which 7 are good. A sample of 3 mangoes is drawn from it.- Find the expected no. of bad mangoes drawn. 6
c) Find Mean and Variance of Geometric distribution. 6
5. a) Given the following bivariate probability distribution obtain :
i) Marginal distributions of X and V
ii) Conditional distribution of X given Y = 2
iii) Condition distribution of Y given X = 0
iv) Expectation of X. 8
0
I
X5
x, x5 x5 X, x5
X,
x,
X,
b) Obtain Mean and Variance of Binomial distribution using cumulant generating
function. 7
x2 + -^ for 0<x<l,0£y £2
0 elsewhere
6. a) If a joint pdf of two dimensional random variable (X.Y) is given by f(x) =
Find : i) P (Y < X) ii) P(Y< % / X<
S
b) Obtain moment generating function of exponential distribution. Hence find
Mean and Variance of the distribution. 7
MT 21: PROBABILITY AND COMBINATORICS (New) (2005 Pattern)
3 Hours
Max. Marks: 70
Instructions: i) Question No, I and Question No. 4 are compulsory, ii) Solve any one from Question Nos. 2 and 3. Hi) Solve any one jmm Question Nos. 5 and 6. iv) Figures to the right indicate full marks.
Solve any four:
a) State and prove the formula for Derangement of n objects.
b) A palindrome is a word thai reads the same from front and backwards, for e.g. LIRIL. How many 7-letter palindromes can be made out of English alphabets ?
In how many ways 5 cakes be given to 7 children, if
i) no child can possess more than one cake.
ii) a child can have any number of cakes.
In how many ways the 6 letters are kept in 6 envelops if two of the letters are too large for one of the envelopes.
Solve the recurrence relation an(? - oa^ + 9an = 0 given that a0 = 5, a. = 9.
b) Show that rC
l) Find how many numbers between 1 and 350 both inclusive are divisible by either 2 or 3 or 5 or 7 ?
-«C + "2c + + nc = n-'cr(l
A(z) =
5-6z + z-
b) There are eight persons. In how many different ways can they be seated around a Round table ?
a) Explain the following terms with examples.
i) Random Variable
ii) Sample Space
iii) Events
r.i <>.
13080] - 205
s/
b) The following table represents the joint probability distribution of the discrete random variable (X, Y). Find all marginal probability distributions and conditional probability distributions of X given Y.
\Y x\ 1 2 — 3
1 1/12 0 1/18
2 1/6 1/9 1/4
3 0 1/5 2/15
c) If a r.v. X is Exponentially distributed then show that
P[X > s + t/X > s] = P[X > t], for any s, t > 0 1
5. a) State Baye's theorem and solve the following. There are 4 boys and 2 girls in
room No. 1 and there are 5 boys and 3 girls in room No. 2. A girl from one of the two rooms laughed loudly. What is the probability the girl who laughed loudly was from room No. 2 ?
b) A random variable X has following: Mx(t) = 2/(2 - t) Find i) E(X) and
ii) Var(X). 8
6. a) Find mean and variance of Hypergeometric distribution. 8
k/v'x ; 0<x<4 0; otherwise
b) The probability density function of continuous r.v. X is given as!
c
f(x) =
Find i) k
ii) Pd < X <2)
iii) Distribution function of X.
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