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ESO 209: PROBABILITY & STATISTICSSemester 2: 2010-11

Assignment #3Instructor: Amit Mitra

[1] Let X be a random variable defined on , P F, . Show that the following are also random

variables; (a) | | X , (b)2

X and (c) X , given that 0 X

.[2] Let 0,1 and F be the Borel field of subsets of . Define X on as follows:

if 0 1 2

1 2 if 1 2 1 X

Show that X defined above is a random variable.[3] Let 1,2,3,4 and 1 , 2,3, 4 F = , , be a field of subsets of . Verify whether

1; X , is a random variable with respect to F .[4] Let a card be selected from an ordinary pack of playing cards. The outcome is one of these 52

cards. Define X on as:

4 if is an ace3 if is a king2 if is a queen1 if is a jack 0 otherwise.

X

Show that X is a random variable. Further, suppose that .P assigns a probability of 1 52 toeach outcome . Derive the distribution function of X .

[5] Let

0 if 1

2 4 if -1 11 if 1.

x

F x x x

x

Show that .F is a distribution function. Sketch the graph of F x and compute the probabilities 1 2 1 2P X , 0P X , 1P X and 1 1 .P X Further, obtainthe decomposition 1d cF x F x F x ; where, d F x and cF x are purelydiscrete and purely continuous distribution functions, respectively.

[6] Do the following functions define distribution functions?

(a) 0, 0

, 0 1 21, 1 2.

x

F x x x

x

; (b) 0, 0

1 , 0 x x

F xe x

; (c) 0 11 1 1.

xF x

x x

[7] Let 330 if 0

2 11 if 03 3

x x

xF x

e e x

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where, x is the largest integer x . Show that .F is a distribution function and compute 6P X , 5P X and 5 8 .P X

[8] The distribution function of a random variable X is given by

2

0, 2,1 3, 2 0,

1 2, 0 5,1 2 5 2, 5 6,1, 6.

x

x

xF x

x x

x

Find 2 5P X , 0 5.5P X and 1.5 5.5 | 2P X X .

[9] Prove that if 1 . ,...., .nF F are n distribution functions, then 1

n

i ii

F x F x is also a

distribution function for any 1,..., n , such that 0i and1

1.n

ii

[10] Suppose 1F and 2F are distribution functions. Verify whether 1 2G x F x F x is also adistribution function.

[11] Find the value of and k so that F given by

2 20 if 0

if 0 x x

F xk e x

is distribution function of a continuous random variable.[12] Let

2

0 if 02 8 if 0 1

2 8 if 1 2

2 8 if 2 31 if 3.

x

x x

F x x x

x c x

x

Find the value of c such that F is a distribution function. Using the obtained value of c , find thedecomposition 1d cF x F x F x ; where, d F x and cF x are purelydiscrete and purely continuous distribution functions, respectively.

[13] Suppose X

F is the distribution function of a random variable X . Determine the distribution

function of (a) X and (b) | | X . Whereif 0

0 if 0 X X

X X

[14] The convolution F of two distribution functions 1F and 2F is defined as follows;

1 2 ;F x F x y dF y x

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and is denoted by 1 2F F F . Show that is F is also a distribution function.[15] Do the following functions define probability mass functions?

(a) 2 2 if 1,2,3,40 otherwise. x x

f x

; (b) ! if 0,1,2,3,4,...0 otherwise.

xe x x f x

where, 0.

(c) ! if 1,2,3,4,...0 otherwise.

xe x x f x

where, 0. [16] Find the value of the constant c such that 1 ; 0,1,2,3... x f x c c x defines a probability

mass function.[17] Let X be a discrete random variable taking values in 3, 2, 1,0,1,2,3 X such that

3 2 1 1 2 3P X P X P X P X P X P X and 0 0 0P X P X P X . Find the distribution function of X .

[18] A battery cell is labeled as good if it works for at least 300 days in a clock, otherwise it is labeled as bad. Three manufacturers, , A B and C make cells with probability of making good cells as 0.95,0.90 and 0.80 respectively. Three identical clocks are selected and cells made by , A B and C areused in clock numbers 1, 2 and 3 respectively. Let X be the total number of clocks working after300 days. Find the probability mass function of X and plot the corresponding distribution function.

[19] Prove that the function 2 if 0

0 otherwise

x xe x f x

defines a probability density function for 0. Find the corresponding distribution function and

hence compute 2 3P X

and 5P X

.[20] Find the value of the constant c such that the following function is a probability density function.

1 if 00 if 0

xc x e x f x

x

where, 0. Obtain the distribution function of the random variable associated with probabilitydensity function f x .

[21] Show that 2 18 if 3 3

0 otherwise x x

f x

defines a probability density function. Find the corresponding distribution function and hence find | | 1P X and 2 9P X