Problem Sheet 4
MA204
Statistics
2008
Department of Mathematics, IIT Madras
1. Show that if X1 , . . . , Xn constitute a random sample from an innite population, then
Cov (Xi X, X ) = 0, for each i = 1, 2, 3, . . . , n.
2. If X1 , . . . , Xn are independe
Problem Sheet 3
MA204
Statistics
2008
Department of Mathematics, IIT Madras
1. Show that if X has the discrete uniform distribution with k values in its range, then its
et (1 ekt )
.
moment generating function is MX (t) =
k (1 et )
2. For the binomial dis
Problem Sheet 2
MA204
Statistics
2008
Department of Mathematics, IIT Madras
1. Find , 2 , and 2 for the random variable X that has the pdf: f (x) = x/2 for 0 < x < 2,
and f (x) = 0, otherwise.
2. Let X be a random variable having mean and variance 2 . Let
Problem Sheet 1
MA204
Statistics
2008
Department of Mathematics, IIT Madras
2x
, for x = 1, 2, 3, . . . , k serve as a probability distribution of a random
k (k + 1)
variable with the given range?
1. Does f (x) =
2. For what values of k the given function
Problem Sheet 0
MA204
Statistics
2008
Department of Mathematics, IIT Madras
1. Show that P (A) + P (B ) 1 P (A B ) P (A) + P (B ).
2. An experiment has ve possible outcomes: A, B, C, D, E. Check whether the following
assignment of probabilities are permis
, ,/'
MA2040 Probability, Statistics and Stochastic Process
Problem Sheet 2
I. For certain binary communication channel, the probability that a transmitted ' 0' is
received as a ' 0' is 0.95 and the probability that a t ransmitted' I ' is received as ' 1
Problem Sheet 5
MA204
Statistics
2008
Department of Mathematics, IIT Madras
1. Show that X is a minimum variance unbiased estimator of the mean of a normal
population. [Compute E ( ln f (X )/)2 ).]
2. Given a random sample of size n from a population with
Problem Sheet 6
MA204
Statistics
2008
Department of Mathematics, IIT Madras
1. Let X1 , X2 constitute a random sample from a normal population with 2 = 1. If the null
hypothesis = 0 is to be rejected in favour of the alternative hypothesis = 1 > 0
whren x
Chapter 10
Elliptic Equations
10.1
Introduction
The mathematical modeling of steady state or equilibrium phenomena generally result
in to elliptic equations. The best example is the steady diusion of heat in any twodomain bounded by . In the absence of an
Dept. of Mathematics, IIT Madras,
MA2020 (July-Nov 2010)
Assignment - 6 (Heat Equation)
Notation:
> 0 is a constant, c2 = K/(), K is the thermal conductivity, is the specic
heat, and is the density of the homogeneous heat conducting material.
1. Solve the
Dept. of Mathematics, IIT Madras,
MA2020 (July-Nov 2010)
Problem Sheet-5 (Wave equation)
Notation: In this problem sheet, c and
used to denote positive constants.
are
Part A - DAlemberts formula
1. Solve
utt 3uxx = 0; x R, t 0,
u(x, 0) = x sin x, x R,
ut
dy
y
y2log x
The given differential equation can be written as -d + - = -x
x
x
Dividing by y2 on both sides and substituting z = 1., we get
Y
dz
z
log x
dx -~ = -~
[1]
Therefore, using the method for non-linear
J
log x dx + c
x2
1 + log x
-+e
x
1 + log x
MA2020 - Dierential Equations
Assignment - 1, July-December 2013
1. Show that each of the following equations is exact and nd a one -parameter family
of solutions.
(i)(3x2 y + 8xy 2 )dx + (x3 + 8x2 y + 12y 2 )dy = 0
(ii) 2xydx + (x2 + y 2 )dy = 0
(iii) co
Assignment Sheet 1
MA2020 Dierential Equations (July - November 2012)
1. Solve the following rst order dierential equations
dy
(a) x dx + y = x3 y 6
dy
(b) xy 2 dx + y 3 = x cos(x)
dy
(c) x dx + y = y 2log(x)
dy
(d) (x2 y 3 + xy) dx = 1
(e)
dy
dx
+ (2x ta
Problem Sheet 7
MA204
Statistics
2008
Department of Mathematics, IIT Madras
1. According to the norms estblished for a reading comprehension test, eigth graders
should average 84.3 with standard deviation 8.6. If 45 randomly selected eigth
graders from a
Probability and Statistics
Test Set 11
1. An electronics manufacturing company produces ICs which have lifetimes
normally distributed with mean (in days) and variance 900 days. Find the
rejection region for testing H0: = 1000 against H1: > 1000 at 1% leve
Probability and Statistics
Test Set 10
1. Let X Bin (1, p), 1/4 p 3/4. Find the MLE and MME of p. Are these
unbiased.
2. Consider a random sample X1,Xn from a double exponential distribution with
density
f (x, , )
1
| x |
expcfw_
, x , R , 0.
Find MMEs
Probability and Statistics
Test Set 9
1. Following are measurements on the nicotine content (in mg) in a random sample
of 72 cigarettes:
1.72
1.24
1.58
1.72
1.74
1.79
1.80
1.68
1.63
1.62
1.73
1.52
1.28
1.54
1.87
1.71
1.87
1.62
1.52
1.23
1.39
1.56
1.79
1.7
Department of Mathematics, IIT Madras
Answer
Quiz-1
MA 204
Statistics
A
Question-cum-Answer Book
Date, Time: Feb 18, 2008, 8:00-8:50 a.m.
Max. Marks: 20
Answer all the questions.
All numerical answers must be in decimals, correct to two decimal places.
Ro
OPERATING SYSTEMS FILE SYSTEMS
Jerry Breecher
10: File Systems
1
FILE SYSTEMS
This material covers Silberschatz Chapters 10 and 11. File System Interface The user level (more visible) portion of the file system. Access methods Directory Structure Protecti
OPERATING SYSTEMS VIRTUAL MEMORY
Jerry Breecher
9: Virtual Memory
1
VIRTUAL MEMORY
WHY VIRTUAL MEMORY? We've previously required the entire logical space of the process to be in memory before the process could run. We will now look at alternatives to this
OPERATING SYSTEMS MEMORY MANAGEMENT
Jerry Breecher
8: Memory Management 1
OPERATING SYSTEM Memory Management
What Is In This Chapter?
Just as processes share the CPU, they also share physical memory. This chapter is about mechanisms for doing that sharing