MSO201A: PROBABILITY & STATISTICS
Problem Set #3
[1] Let X be a random variable defined on(Ω, ℱ, 𝒫) . Show that the following are also random variables;
(a) | X |, (b) X 2 and (c) X , given that { X < 0} = f .
s - field of subsets of Ω. Define X
if 0 £ w £ 1 2
ìw
X (w ) = í
îw - 1 2 if 1 2 < w £ 1
[2] Let Ω = [0,1] and ℱ be the Borel
on Ω as follows:
Show that X defined above is a random variable.
[3] Let Ω = {1,2,3,4} and ℱ = 1𝜙, Ω, {1}, {2,3,4}3 be a s - field of subsets of Ω. Verify whether
X (w ) = w + 1; "w Î W , is a random variable with respect to ℱ.
[4] Let a card be selected from an ordinary pack of playing cards. The outcome
cards. Define X on Ω as:
ì4
ï3
ïï
X (w ) = í2
ï1
ï
ïî0
w
is one of these 52
if w is an ace
if w is a king
if w is a queen
if w is a jack
otherwise.
Show that X is a random variable. Further, suppose that P (.) assigns a probability of 1 52 to each
outcome w . Derive the distribution function of X .
if x < -1
ì0
ï
[5] Let F ( x ) = í( x + 2 ) 4 if -1 £ x < 1
ï1
if x ³ 1.
î
Show that F (.) is a distribution function. Sketch the graph of F ( x ) and compute the probabilities
Further, obtain the
P ( -1 2 < X £ 1 2 ) , P ( X = 0 ) , P ( X = 1) and P ( -1 £ X < 1) .
decomposition F ( x ) = a Fd ( x ) + (1 - a ) Fc ( x ) ; where, Fd ( x ) and Fc ( x ) are purely discrete
and purely continuous distribution functions, respectively.
[6] Which of the following functions is(are) distribution functions?
ì0, x < 0
x<0
ìï0,
ì0
ï
(a) F ( x ) = í x, 0 £ x £ 1 2 ; (b) F ( x ) = í
; (c) F ( x ) = í
-x
ïî1 - e , x ³ 0
î1 - 1 x
ï1, x > 1 2.
î
x £1
x > 1.
[8] The distribution function of a random variable X is given by
Page
P ( X > 6 ) , P ( X = 5 ) and P ( 5 £ X £ 8 ) .
1
if x £ 0
ì0
ï
[7] Let F ( x ) = í
2 - x 3 1 -[ x 3]
- e
if x > 0
ïî1 - 3 e
3
where, [ x ] is the largest integer £ x . Show that F (.) is a distribution function and compute
x < -2,
ì0,
ï1 3,
-2 £ x < 0,
ï
ï
0 £ x < 5,
F ( x ) = í1 2,
2
ï
1
2
+
x
5
2, 5 £ x < 6,
(
)
ï
ïî1,
x ³ 6.
Find P ( -2 £ X < 5 ) , P ( 0 < X < 5.5 ) and P (1.5 < X £ 5.5 | X > 2 ).
n
[9] Prove that if F1 (.) ,...., Fn (.) are n distribution functions, then F ( x ) = å a i Fi ( x ) is also a
i =1
distribution function for any
(a1,...,a n ) , such that ai ³ 0 and
n
å a i = 1.
i =1
[10] Suppose F1 and F2 are distribution functions. Verify whether G ( x ) = F1 ( x ) + F2 ( x ) is also a
distribution function.
[11] Find the value of a and k so that F given by
ìï0
F ( x) = í
- x2
ïîa + k e
if x £ 0
2
if x > 0
is distribution function of a continuous random variable.
if x < 0
ì0
ï x + 2 8 if 0 £ x < 1
)
ï(
ï
[12] Let F ( x ) = í x 2 + 2 8 if 1 £ x < 2
ï
ï( 2 x + c ) 8 if 2 £ x £ 3
ïî1
if x > 3.
Find the value of c such that F is a distribution function. Using the obtained value of c , find the
decomposition F ( x ) = a Fd ( x ) + (1 - a ) Fc ( x ) ; where, Fd ( x ) and Fc ( x ) are purely discrete
(
)
and purely continuous distribution functions, respectively.
[13] Suppose FX is the distribution function of a random variable X . Determine the distribution function
of (a) X + and (b) | X |. Where,
ìX
X+ =í
î0
if X ³ 0
if X < 0
[14] The convolution F of two distribution functions F1 and F2 is defined as follows;
#
)
x! if x = 0,1,2,3,4,...
otherwise.
Page
(
ìï e - l l x
ì( x - 2 ) 2 if x = 1,2,3,4
(a) f ( x ) = í
; (b) f ( x ) = í
0
otherwise.
î
ïî0
2
★
𝐹(𝑥) = ∫$# 𝐹! (𝑥 − 𝑦) 𝑑𝐹" (𝑦) ; 𝑥 ∈ ℛ,
and is denoted by F = F1 F2 . Show that F is also a distribution function.
[15] Which of the following functions are probability mass functions?
(
ìï e - l l x
(c) f ( x ) = í
ïî0
where, l > 0.
)
where,
l > 0.
x! if x = 1,2,3,4,...
otherwise.
[16] Find the value of the constant c such that f ( x ) = (1 - c ) c x ; x = 0,1,2,3... defines a probability
mass function.
[17] Let X be a discrete random variable taking values in X = {-3, -2, -1,0,1,2,3} such that
P ( X = -3) = P ( X = -2 ) = P ( X = -1) = P ( X = 1) = P ( X = 2 ) = P ( X = 3)
P ( X < 0 ) = P ( X = 0 ) = P ( X > 0 ) Find the distribution function of X .
and
[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 are
used in clock numbers 1, 2 and 3 respectively. Let X be the total number of clocks working after
300 days. Find the probability mass function of X and plot the corresponding distribution function.
ìq 2 x e - q x
[19] Prove that the function fq ( x ) = í
if x > 0
î0
otherwise
defines a probability density function for q > 0. Find the corresponding distribution function and
hence compute P ( 2 < X < 3) and P ( X > 5 ) .
[20] Find the value of the constant c such that the following function is a probability density function.
ìc ( x + 1) e - l x if x ³ 0
fl ( x ) = í
if x < 0
î0
where, l > 0. Obtain the distribution function of the random variable associated with probability
density function f l ( x ) .
ì x 2 18 if - 3 < x < 3
[21] Show that f ( x ) = í
otherwise
î0
defines a probability density function. Find the corresponding distribution function and hence find
3
)
Page
(
P (| X | < 1) and P X 2 < 9