Sample problems.
Find the indenite integrals of the following functions.
1.
x sin x dx
2.
x2 ex dx
1 x2 dx
3.
log log x
x
4.
dx
5.
(sin x)3 (cos x)6 dx
6.
(sin x)2 (cos x)2 dx
7.
esin x cos x dx
8.
1
x2 +9 dx
9.
1
x2 9 dx
10.
x arctan x dx
Calculate the f
Final Examination.
Due Dec 15.
1. We have a population of size N which completely renews itself every generation. The
total size is N in every generation. The population consists of two types of individuals
A and B . If in a given generation the populatio
Assignment 9.
1. Let us consider independent random variables Xi = 1 with probability
1
2
each.
Sn = X1 + X2 + + Xn
and
j
Sj
n ( ) =
n
n
n is interpolated linearly between
j
n
and
j +1
.
n
Prove the estimate
E [ | n ( t) n ( s ) | 4 ] C | t s | 2
for the
Probability and Statistics
Home Work due March 31, 2005.
Q1. X is a random variable having a Normal distribution with mean 0 and unknown
variance 2 . Test the null hypothesis that 2 = 1 against the alternative 2 = 2 at 5%
level of signicance based on a si
The rst part was done in class. Second part is Homework.
1. The number of accidents involving one or more fatalities in a stretch of Interstate 80
were recorded for a year and the data is as follows. Is the data consistent with a Poisson
distribution for
Probability and Statistics
Home Work due March 3, 2005.
Q1. If X is a random variable having a normal distribution with mean 0 and variance 1,
i.e. having the density
x2
1
f (x) = e 2
2
show that the probability density of the random variable Y = X 2 is g
Probability and Statistics
Home Work due Feb 24, 2005.
Q1. If X1 and X2 are independent and uniformly distributed in [0, 1] what is the probability distribution of Y = X1 X2 ?
Q2. If X1 and X2 are independent and have the exponential distribution with den
Home work due Feb 17, 2005.
Q1. Let X be a random variable whose distribution is given by the probability density
f (x) =
1 |x| if |x| 1
0
if |x| > 1
Calculate E [X ], E [|X |] and V (X ).
Q2. X, Y are two random variables with a joint distribution given
Probability and Statistics
Home Work due Feb 10, 2005.
From a bag containing 10 bills each of denominations $1, $5 and $10, a bill is drawn at
random. Then another bill is drawn with out replacement. Let X1 be the face value of
the rst bill and X2 the fac
Probability and Statistics
Home Work due Feb 3, 2005.
Q1. A random experiment consists of drawing a card from an ordinary deck of 52 cards.
1
Each card is assigned the probability 52 . C1 is the event (set) consisting of all thirteen
hearts and C2 is the
Dierential equations
A dierential equation is a relationship between a function and its derivatives.
Examples: f (x) is the function f (x), f (x), are the derivatives. A dierential equation
is a relation of the form (f, f , f , ) = 0.
1. f (x) = sin x
2.
Nov 9 and 11, 2009.
Conic Sections
Conic sections consist of three classes of curves. Ellipses, Hyperbolas and Parabolas.
The equation
y2
x2
+ 2 =1
a2
b
represents an ellipse. It is like an oval, with an axis along the x axis of length 2a and an
axis alon