# Math 411 — Problem Set 6 Issued: 10.17 Due: 10.24 ```Texas A&amp;M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2011
Math 411 — Problem Set 6
Issued: 10.17
Due: 10.24
6.1. A group of 20 people go out to dinner. 10 go to an Italian restaurant,
6 to a Japanese, and 4 to a French restaurant. The fractions of people
satisfied with their meals were 4/5, 2/3, and 1/2 respectively. The next
day the person you a talking to was satisfied with what they ate. What
is the probability that they went to the Italian restaurant?
6.2. Fill in the rest of the joint distribution so that X and Y are independent. There are two possible answers.
X=0 X=1
Y =0
?
2/9
Y = 1 2/9
?
6.3. Consider f (x) = c(1 − x2 ) for −1 &lt; x &lt; 1 and 0 otherwise. What
value of c should we take to make f a density function?
6.4. Suppose X has density function x−2/3 /21 for 1 &lt; x &lt; 8 and 0 otherwise. Find EX, E(X 2 ), and var(X).
6.5. Suppose X has density function F (x) = x/2 for 0 &lt; x &lt; 2 and 0
otherwise. Find P (X &lt; 1) and P (X &gt; 3/2).
6.6. Suppose X has density function f (x) = 4x3 for 0 &lt; x &lt; 1 and 0
otherwise. Find (a) the distribution function F (x) = P (X ≤ x), (b)
P (X &lt; 1/2), and (c) P (1/3 &lt; X &lt; 2/3).
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