emon87
12-04-2006, 10:51 PM
Hello,
These are some extra credit questions from my Probability/Stats class. Can you guys do them? I struggled with them, but I think I got them in the end.
Problem 1 (25 points). Find the cumulative distribution function FX and the density fX of a
continuous random variable X with the property that
P(X > 2x) = (P(X > x))^2 for all x in R,
fX(0+) = lim
x!0+
fX(x) and fX(0−) = lim
x!0−
f(x) exist (i.e are finite numbers).
Give an example of a random variable such that fX(0+) = limx!0+ fX(x) does not exist.
Problem 2 (25 points). Assume 7 numbers are chosen independently at random from {1, 2, 3, . . . , 286}.
Find the probability that at least 2 of these 7 numbers have the same remainder at division by either
11 or 13.
The same problem, only that the numbers are chosen from {1, 2, 3, . . . , 287}.
Problem 3 (25 points). There people A,B,C are playing the following game. They pick a card
with replacement at random form a deck of 52 cards, and then one player wins 1$ only if his card
is greater than the cards of the others, otherwise the game is undecided. Find the probability that A
wins the game.
Another variant of this game is the following. Each player is given a number at random from
{1, 2, . . . , n} and one wins if his number is greater than the others. Find the probability that A wins.
What happens with this probability when n tends to 1? How about if we have 100 players?
Problem 4 (25 points). In the Lotto problem one picks 6 numbers from {1, 2, . . . , 42}. Then 6
numbers are drawn at random without replacement. One can win with 3/6, 4/6, 5/6 or 6/6 if either
ticket wins accordingly. We call a win if the player matches at least 3 of the drawn numbers. In
class we discussed the chances of winning with one ticket bought.
Assume the player buys two tickets. What is the probability of winning then? What is best
strategy to win with 3/6, 4/6, 5/6, 6/6? If there are common numbers on both tickets does this
improve your chance of winning with any of the 3/6, 4/6, 5/6 or 6/6?
How about if you buy three tickets or more? What is the probability of winning? Which strategy
is going to give better results?
If you can do them, that would be awesome so I can check my answers.
These are some extra credit questions from my Probability/Stats class. Can you guys do them? I struggled with them, but I think I got them in the end.
Problem 1 (25 points). Find the cumulative distribution function FX and the density fX of a
continuous random variable X with the property that
P(X > 2x) = (P(X > x))^2 for all x in R,
fX(0+) = lim
x!0+
fX(x) and fX(0−) = lim
x!0−
f(x) exist (i.e are finite numbers).
Give an example of a random variable such that fX(0+) = limx!0+ fX(x) does not exist.
Problem 2 (25 points). Assume 7 numbers are chosen independently at random from {1, 2, 3, . . . , 286}.
Find the probability that at least 2 of these 7 numbers have the same remainder at division by either
11 or 13.
The same problem, only that the numbers are chosen from {1, 2, 3, . . . , 287}.
Problem 3 (25 points). There people A,B,C are playing the following game. They pick a card
with replacement at random form a deck of 52 cards, and then one player wins 1$ only if his card
is greater than the cards of the others, otherwise the game is undecided. Find the probability that A
wins the game.
Another variant of this game is the following. Each player is given a number at random from
{1, 2, . . . , n} and one wins if his number is greater than the others. Find the probability that A wins.
What happens with this probability when n tends to 1? How about if we have 100 players?
Problem 4 (25 points). In the Lotto problem one picks 6 numbers from {1, 2, . . . , 42}. Then 6
numbers are drawn at random without replacement. One can win with 3/6, 4/6, 5/6 or 6/6 if either
ticket wins accordingly. We call a win if the player matches at least 3 of the drawn numbers. In
class we discussed the chances of winning with one ticket bought.
Assume the player buys two tickets. What is the probability of winning then? What is best
strategy to win with 3/6, 4/6, 5/6, 6/6? If there are common numbers on both tickets does this
improve your chance of winning with any of the 3/6, 4/6, 5/6 or 6/6?
How about if you buy three tickets or more? What is the probability of winning? Which strategy
is going to give better results?
If you can do them, that would be awesome so I can check my answers.