MATH 255 Midterm1Solutions.pdf.pdf

Matn 255
5?““3 2014 -20
M.‘Jl—&rm 1~ AV\SWUS
All numeric answers must be simplified to a real numbe
r or a fraction of two integers
with no common factors. Show your work legibly
to maximize partial credit.


Problem 1. [6 pts] Consider a probability space (@, P) and
let 4, B,C C Q be three events with
P(A)=0.1, P(B) = 0.2, and P(C) = 0.3. (a) Compute P[(AU
B)C|B*UC] under the assumption
that A, B, C are independent. (b) Assume instead that (i) A, B, and C are pairwise indepe
ndent;
and (ii) B and C are conditionally independent given A. Compute P(A|BUC). (Each part 3
points.)


P[(AUB)C|BUC] = %
25 PABUC) = 5. 1
Problem 2. [6 pts] An urn contains only white balls,
while another urn contains 30 white and 10
black balls. One of the two urns is selected at random
and then a ball is drawn (at random) from
that urn. The ball turns out to be white (call this event
W;) and is then put back into the urn.
What is the conditional probability that another ball drawn
from the same urn will be black (call
this event By)?



P(Bszl) = 3/2—8



Problem 3. [6 pts] A fair coin is tossed repeatedly and
independently until two consecutive heads
or two consecutive tails appear. Let X be the durati
on of this game (number of coin tosses).
Compute the PMF px(4) and the expectation E[X]. (Each part 3
points.)


px(4) = %3 EX]= 3

Problem 4. [6 pts] Let X and Y denote the number of 1s and 6s,
respectively, that turn up inn
independent throws of a fair die with 6 faces. Compute E[XY].


E[XY] = A (“_‘)/3é

Problem 5. [6 pts] Let X and Y be independent Poisso
n random variables with parameters \; and
Ay, respectively. Let Z = X + Y. Determine the PMFE
pz(k) for all k > 0. Compute E[X|Z =n)
as afunction of n. (Each part 3 points.)


\ k —()x"’),)
pz(lc)=(/\"Jr 3> € E[X|Z =n] = M._L :
k)
31"‘%\7

]A:O,'\/'L/\\;

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) (V 19 — (2 )¢ T [ -
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CRENRCT
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f¢g=(2724"' 2e=(9)d bo=1(v)d (‘1)

-_— 50 e 8&9
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