Introduction to Probability
Math 30530, Section 01 Fall 2012
Homework 9 Solutions
1. During a season, a basketball team plays 70 home games and 60 away games. The
coach estimates at the beginning of the season that the team will win each home
game with pr
Introduction to Probability
Math 30530, Section 01 Fall 2012
Homework 6 Solutions
1. GW 17.2
Solution: a) 0, 1, 2, 3, . . . Its Poisson because were counting number of occurrences of an event, with dierent occurrences independent. We have = 6.85.
b) .1487
a 5
@MXU) = E(etX) = Zetxp(x) 2 £03: +62: +83: +e4r +£51).
1:1
,h
3. Note that
no 1 x 00 00 .
MXO) _ E(etX) 2 Zen . _._. 2:81: . e~xln3 : 226104;.-
x=l x=l x=l
Restricting the domain of Mx(t) to the set {1: t < ln3} and using the geometric series
theore
Introduction to Probability
Math 30530, Section 01 Fall 2012
Homework 2 Solutions
1. For events A and B with Pr(A) > 0, Pr(B) > 0, we say that A is positively correlated
with B if Pr(A|B) > Pr(A) (in other words, B occurring increases the chance that A
oc
Introduction to Probability
Math 30530, Section 01 Fall 2012
Homework 4 Solutions
1. GW 8.2
Solution: For x 1, y 1 (and integers),
Pr(X = x, Y = y) = (.99)x1 (.01)(.97)y1 (.03)
(here using independence of X and Y ). For all other values of x, y, Pr(X = x,
Introduction to Probability
Math 30530, Section 01 Fall 2012
Homework 1 Solutions
1. A box contains four candy bars: two Mars bars, a Snickers and a Kit-Kat. I randomly
draw a bar from the box and eat it, then draw a second and eat that, too. I record the
Introduction to Probability
Math 30530, Section 01 Fall 2012
Homework 3 Solutions
1. You are in a room with ve doors, and your host tells you that behind two randomly
chosen doors he has placed a prize (all choices of two doors equally likely). You open
t
I CDMWWWO=UWBNJLM®M=Wmf+NWpnzwm
py(0) = 5/25, and p(l, O) = 1/25. Since [20, 0) aé px(l)py(0), X and Y are dependent.
Magma?er
By the independence of X and Y, ' ~r .
P(X = 1, Y = 3) = P(X =1)P(Y = 3) = - = g}.
P(X+Y=3)=P(X=1,Y:2)+P(X=2,Y=1)
1 2 1 2 2
Introduction to Probability
Math 30530, Section 01 Fall 2012
Homework 5 Solutions
1. GW 15.7
a) If X is number of draws, then X Geometric(4/52), so E(X) = 52/4 = 13.
b) Taking at least four is the same as failing on the rst three trials, so the probabilit
Introduction to Probability
Math 30530, Section 01 Fall 2012
Homework 8 Solutions
1. GW 29.1a): The joint density is 2/9 on the triangle (and 0 elsewhere).
E(X + Y ) =
triangle
2. GW 29.4:
2
(x + y) dA =
9
3
x=0
3x
y=0
2
(x + y) dydx = 2.
9
x
1
y 18e2x7y
Hangover/5K (C5 chalx'ms
@ Let f be the probability density function of a gamma random variable with parameters r and
A. Then XXVI?
f (x) '- "56?"
Therefore,
I A, r - x r-2 _, Ar+l r2 M . r 1
f(x)=r(r)[_le Axx 1+ek(r_1)x ]_F(r)x e (x A
This relati
Introduction to Probability
Math 30530, Section 01 Fall 2012
Homework 7 Solutions
1. GW 25.1: The area of the triangle is 9/2, so on the triangle the joint density is
2/9; everywhere else it is 0. The region R of the triangle where X + Y > 2 is shown
shad