Name
ID#:
EL~630
Quiz # 5
(a) Let X be an exponential random variable with parameter it.
Dene
Y = .X + 3m
X
Determine the probabiiity density function (p.d.f) of Y.
(b) Find the p.d.f of Y = cos2(:a¥), where
fX(x)={2x 0<x<1
0, otherwise. Name
tD#:
EL-630 Probability
Quiz #1
A box contains a white balls, b black balls and c red balls. Balls are
drawn at random without replacement from the box one after the other.
a) What is the probability that the first ball drawn is white?
b) What is the probabili
EL-630 Probability
Quiz #2
1. a) X is an exponential random variable with parameter = 1. Thus
x0
e x ,
f X ( x) =
0, otherwise
Determine the event A = ( X 1) ( X > 10). Find f X ( x | A).
Plot it.
Solution: P ( A) = P[( X 1) ( X > 10)] = P ( X 1) + P (
EL630 - Solutions to HW #1
1. Note that A, B, mutually exclusive implies A B = and hence P ( A B) = . Further
A, B are independent
P ( A B ) = P( A) P( B )
Let the pair (fi, sk) represent the elementary event outcome on first roll is i and second roll
is
EL630 - Solutions to HW #2
1. Let Ai = switch Si is closed, for i =1,2,3,4. Then P(Ai) = p; i = 1,2,3,4 (given) and
P( Aj ) =1-p. Moreover P(Ai Aj ) = P(Ai) P(Aj) etc (i j ) because of the independence of events
(given) now from the diagram a signal will
EL-630 Solutions to HW #4
1. In any roll, p = the probability that two appears = 1/6. Hence q = probability that two does not appear = 1 p = 5/6. X = number of rolls required to get a two. You may get the two on the first shot (X = 1) or the second shot (
EL-630 Solutions to HW #4
1. In any roll, p = the probability that two appears = 1/6. Hence q = probability that two does not appear = 1 p = 5/6. X = number of rolls required to get a two. You may get the two on the first shot (X = 1) or the second shot (
EL-630 Solutions of HW #5
1.
1/ 5,
f X ( x) =
0,
f X ( x)
3< x < 2
otherwise.
3
1
1
2
x
(a) Pcfw_ X 2 > 1 = Pcfw_1 > X > 1 = Pcfw_ X shaded area above
= Pcfw_(3 X < 1) (1 < X 2)
Disjoint events
1
2
= Pcfw_3 X < 1 + Pcfw_1 < X 2 = 3 f X ( x) dx + 1 f X (
EL 630 - Homework #1
1. A fair die is rolled twice. For this experiment, define events A and B such that the events are
(a) disjoint (mutually exclusive), but not independent
(b) independent but not mutually exclusive
(c) independent and mutually exclusiv
Homework #2
1. The four switches in the figure operate independently. Each switch is closed with probability
p and opened with probability (1-p).
(a) Find the probability that a signal at the input will be received at the output.
(b) Find the conditional
EL 630 - HW #3
1. A fair coin is tossed 4 times. The random variable X is defined as the total number of heads.
Write the events cfw_ X = 3, cfw_ X 2, cfw_ X < 2, cfw_ X = 5.
2. Do the following functions define probability distribution functions?
1 e x ,
EL 630 - Homework #4
1. A fair die is rolled until a two appears. Let the random variable X be the number of rolls
required. Find the probability distribution function FX(x) of the random variable X.
2. A fair coin is rolled 5,000 times. What is the proba
EL 630 - Homework #5
1. The p.d.f of a continuous random variable X is given by
1/ 5, 3 < x < 2,
f X ( x) =
0, elsewhere
Find (a ) P ( X 2 > 1) (b) P ( sin( X 0 ) .
2. The probability of heads p of a random variable is itself a random variable X (i.e.,
EL 630 - Homework #6
1. (a) X is a Gaussian (or Normal) random variable with zero mean and variance 2 . Define
Y = aX + . Find the characteristic function of Y (Y (u ).)
(b) X ~ N(0, 2) (as before). Define Z = X2. Find the mean, variance and characteristi
EL-630: Probability Theory
Midterm Exam, March 7, 2001.
(Answer all problems, All problems carry equal weights)
1(a) Let A and B be two events such that P (A) = p1 > 0 P (B ) = p2 > 0 p1 +p2 > 1.
Show that
P (AjB ) 1 ; (1 ; p1) :
p
2
(b) A biased coin wit
7 E e EG I B E P B C Q H E Q P U 9 Q P B E V U CS H E WG P E E B Q P P H 9 E S P 9 HG E e E G I B E V U CS E Q P P 9 Q P `PGSG t 9 t C B I E Q P F U G u @ t8 q s o s p r r o @ o 6t 6 8 q @ o8 n ` t U E D G V H G m R C U C G P e U c R ` P G H U E F ` P G S
EL-630 Solutions to HW #6
1. (a) X ~ N(0, 2). Then its characteristic function is
X ( ) = Ecfw_e jXu = e u
22
/2
we need Y (u ) where Y = aX +
Y (u ) = Ecfw_e jYu = Ecfw_e j ( aX + ) u = Ecfw_e ju Ecfw_e jaXu = e ju Ecfw_e jaXu
= e ju X (au ) = e j