B.Sc-B.Sc Statistics 2nd Sem Discrete Probability and Probability Distributions(University of Pune, Pune-2013)

                                                            F.Y. B.Sc.

SEAT No. :

[Total No. of Pages : 4

STATISTICS/STATISTICALTECHNIQUES

Discrete Probability and Probability Distributions

(2008 Pattern) (Paper - II)

Time : 3 Hours]                                                                      [Max. Marks : 80

Instructions to the candidates :

1) All questions are compulsory.

2) Figures to the right indicate full marks.

3) Use of statistical tables and calculator is allowed.

4) Symbols have their usual meanings.

Q1) a) Choose correct alternative for the following :

[4 × 1 = 4]

i)        The probability that there are 53 Mondays in a randomly chosen

leap year is

A) 1/7                                                     B) 1/14

C) 1/28                                                   D) 2/7

ii)       For a sample space    = {w1, w2, w3, w4}, P (w1) = p, P (w2) = 2p,

P (w3) = 3p, P (w4) = 4p, where p > 0. For what value of p will this

be a probability model?

A) 1/100                                                B) – 1 C) 10             D) 1/10

iii)      The probability mass function (p.m.f) of a discrete random variable

(r.v.) X is given by,

X                      1        2         3        4       5

P ( X = x)       0.1 0.25 0.25 0.2 0.2

What is the P (2 < X < 5)?

A) 0.9                                                     B) 0.5

C) 0.45                                                   D) 0.3

iv)      If X and Y are any two r.vs, then the covariance between a X + b

and cY + d (where a, b, c, d are constants) is :

A) Cov (X, Y)                                     B) abcd Cov (X, Y)

C) ac Cov (X, Y)                                D) ac Cov (X, Y) + bd

b) State whether the following statements are true or false : [4 × 1 = 4]

i) An event containing all the points of sample space is called as an

impossible event.

P.T.O.

1. If E (Y) = 3 where Y = x- 2 , then E (X) = 17.

5

iii) The first raw moment of a variable is always variance.

iv) Mode of binomial distribution is unique.

c) Define finite sample space and give one example of it.                       [2]

d) If the probability generating function (p.g.f.) of a r.v.X is given by

G x (t)=⎜⎛ 2 + 1 t⎞10 . Find the distribution of X. ⎟

⎝3 3⎠

e) Define conditional probability of an event.

f) Let A, B, C be any three events defined on

following events.

i) At least one event occurs,                    

ii)       Exactly one event occurs.

Q2) Attempt any four of the following :                                                 [4 × 4 = 16]

a) Define the following terms :

i) Event,                                                         

ii) Mutually exclusive events,

iii) Complement of an event,                  

iv) Independence of two events.

b) If A and B are any two events defined on , then prove that,

P (A 8 B) = P (A) + P (B) – P (A 1 B)

c) If P (A) = 0.6, P (B) = 0.5, P (A 1 B) = 0.3

Compute :

i) P (Exactly one of A and B occurs)

ii) P (A'/B), where A' is the complement of A.

d) State and prove Bayes’ theorem.

e) If X and Y are two discrete random variables then, prove that

Var (aX + bY) = a2 Var (X) + b2 Var (Y) + 2ab Cov (X, Y), where a and

b are real constants.

f) State axioms of probability. For an event A defined on , prove that

P (A') = 1 – P (A).

Q3) Attempt any four of the following :                                                 [4 × 4 = 16]

a) Let A, B, C be three events defined on      such that

[4317]-14

P (A) = 0.3, P (B) = 0.2, P (C) = 0.5, P (A 1 C) = 0 = P (B 1 C), P (A 1 B) = 0.17.

Calculate :

i) P (A 8 B 8 C).

ii) P (A' 1 B' 1 C').

2

b) A bag contains 4 tickets numbered 446, 464, 644 and 666. One ticket is

drawn randomly. Let Ai (i = 1, 2, 3) be the event that the ith digit of the

number of the ticket is 4. Discuss independence of the events A1, A2, A3.

c) Following is the probability distribution of a discrete r.v.X.

X                   12345

P [X = x] 3k 5k 2k k k

Find

i) The value of k,

ii) P [X is even],

iii) Mode, iv) E (X).

d) Define cumulative distribution function of a discrete r.v. and state any

two properties of it.

e) The p.m.f of a.r.v. X is

P (X = x) = x + 2 ,for x =1,2,3,4,5.

25

= 0, otherwise

Find E (X) and V (X). Using these results find the values of

i) V (– 3X + 4).⎠

1. ® B (n1, p); Y ® B (n2, p), if X and Y are independent. Find the

conditional probability distribution of X given X + Y = n. Identify the

distribution.

Q4) Attempt any two of the following :                                                  [2 × 8 = 16]

a) i)         Define :

1) A bivariate discrete r.v. and

2) Joint probability mass function of (X, Y).

ii)       The probability distribution of a discrete r.v.X is given below :

X                 0123

P [X = x] 0.1 0.3 0.4 0.2

Find the third central momentm3. Also comment on the nature of

the distribution.

[4317]-14                                                         3

b) Let (X, Y) be a bivariate r.v. with the following joint p.m.f.

(x, y) (0, – 1) (0, 1) (1, – 1) (1, 1)

P (x, y) 2/25          3/25         8/25 12/25

Find

i) E (X) and E (Y), ii) V (X) and V (Y),

iii) Cov (X, Y) and

iv) Correlation coefficient between X and Y.

1. ® B (n, p). Obtain the mean and variance of X. Show that Var
2. ® P (m). If P (X = 2) = P (X = 1), find P (X > 4).

(X) < E (X).

ii) Find recurrence relation between the probabilities of Poisson

distribution with parameter m.

Q5) Attempt any two of the following :                                                  [2 × 8 = 16]

a) Define a discrete uniform probability distribution. Give one real life

situation where it is applicable. Also, computeg1 for a discrete uniform

r.v. and comment upon its value.

b) i) Show that all raw moments of a Bernoulli (p) r.v. are equal to ‘p’.

1. ® B (5, 1/2)

and Y® B (8, 1/2).

Find P⎛ x + y³1⎞

⎝⎜2

c) Define Hypergeometric distribution. Give one real life situation where it

can be used. Also obtain its mean.

d) The joint p.m.f. of (X, Y) is P (x, y) = 2x + 3y , 72

                  x =0,1, 2;

               y = 1, 2, 3

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