Indian Institute of Information Technology Allahabad
Probability and Statistics
Problem Set 05
1. Check weather the following functions are distribution functions of 2-dim random vector or not.
1 if x + 2y 1
(a) F (x, y) =
0 if x + 2y < 1.
0 if x < 0 or x
Module 2
Random Variable and Its Distribution
1. Random Variable
Let , , be a probability space. On many occasions we may not be directly interested in
the whole sample space . Rather we may be interested in some numerical characteristic of the
sample spa
Module 1
Probability
1. Introduction
In our daily life we come across many processes whose nature cannot be predicted in advance.
Such processes are referred to as random processes. The only way to derive information about
random processes is to conduct e
Indian Institute of Information Technology Allahabad
Probability and Statistics
Problem Set 01
1. Let be a sample space of a random experiment and F be a -field (or -algebra) on the subset of . Let
A, B, C F. What is the event that only A occurs? What is
Indian Institute of Information Technology Allahabad
Probability and Statistics
Problem Set 03
1. Twenty distinguishable balls are placed at random in six boxes that are labeled as B1 , . . . , B6 . Find the
probability that boxes with labels B1 , B2 , an
Module 3
Function of a Random Variable and its distribution
1. Function of a Random Variable
Let , , be a probability space and let be random variable defined on , , . Further
let : be a given function and let : be a function of random variable , defined
Indian Institute of Information Technology Allahabad
Probability and Statistics
Problem Set 04
1. Let X Bin(n, p), where n is a positive integer and p (0, 1). Let Y1 = n X, Y2 = X 2 , and Y3 =
Show that Y1 , Y2 , and Y3 are random variables and find the d
Data Structure Lab Program - 2016
Lab exercise 9
Note: All programs must be written in C following coding rules as mentioned on the course website.
1.
Read n ints and make a binary search tree (BST). Do k search operations to print results as y/n.
Input:
Indian Institute of Information Technology Allahabad
Probability and Statistics
Answers of Problem Set 01
1. (a) A B c C c , (b) (A B) (A C) (B C), (c) A B C c , (d) (Ac B c ) (Ac C c ) (B c C c ).
2. (a) No, (b) No, (c) Yes.
3. (a) True, (b) False.
4. (a
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Probability and Statistics
Hints/Solutions to Assignment 5
1. P(1 < X + Y < 2) = 2 e-1 3 e-2. P( X < Y | X < 2Y ) = ,
P(0 < X < 1|Y = 2) = 1 e-1. P(X + Y < m) = , gives m as a solution of
2(m+1) = em, or approximately 1.6784.
2. The marginal distributions
Probability and Statistics
Hints/Solutions to Assignment No. 4
1.
The range of Y is 0 to 4. For 0 y 1, there are two inverse images
y . For 1 < y < 4, there is one inverse image
density of Y as
2 1/2
f Y (y)
y,
0 y 1,
=
9
1
(1 + y 1/2 ), 1 < y < 4,
=
9
e
Probability and Statistics
Assignment No. 6
1. Let (X,Y) have the joint pmf
Y\ X
-2
1
2
-1
1/6
1/6
1/12
0
1/12
1/12
0
1
1/6
1/6
1/12
Find the joint pmf of (U,V) where U = X and V = Y2.
2. Projectiles are fired at the origin of an XY coordinate system. Ass
Probability and Statistics
Assignment No. 5
1. Let (X,Y) have the joint pdf f(x,y) = e-(x+y) , x > 0, y > 0. Find P(1 < X+Y < 2),
P(X < YX < 2Y), P(0 < X < 1Y = 2). Determine such that P(X+Y < m) = .
2. Let (X,Y) have the joint pdf f(x,y) = x+y, 0 < x < 1
Probability and Statistics
Assignment No. 4
1. Let X be a continuous random variable with density function given by
fX(x) = 2(x + 1)/9 ,
-1 < x < 2
= 0,
otherwise.
Find the density function of Y = X2.
2. Let X be continuous random variable with density gi