Okay so these aren't all Permutations and Combinations but I've been making my way through the first half of this handy resource recently http://ift.tt/1z0IN87 and there are 5 questions I haven't been able to figure out on my own. Any advice would be appreciated. I have some notes at the bottom to explain what difficulty I'm having.
1) An urn contains 10 marbles in which 3 are black. Four of the marbles are
selected at random (without replacement) and are tested for the black color.
Define the random variable X to be the number of the selected marbles that
are not black.
(a) Find the probability mass function of X.
(b) What is the cumulative distribution function of X?
(c) Find the expected value of X.
2) A box contains 7 marbles of which 3 are red and 4 are blue. Randomly select
two marbles without replacement. If the marbles are of the same color then
you win $2, otherwise you lose $1. Let X be the random variable representing
your net winnings.
(a) Find the probability mass function of X.
(b) Compute E(2X)
3) Suppose that X is a discrete random variable with probability mass function
p(x) = cx2
, x = 1, 2, 3, 4.
(a) Find the value of c.
(b) Find E(X) and E(X(X − 1)).
(c) Find Var(X).
4) A box contains 3 red and 4 blue marbles. Two marbles are randomly selected
without replacement. If they are the same color then you win $2. If they are
of different colors then you lose $ 1. Let X denote the amount you win.
(a) Find the probability mass function of X.
(b) Compute E(X) and E(X^2).
(c) Find Var(X).
5) The probability of a student passing an exam is 0.2. Ten students took the
exam.
(a) What is the probability that at least two students passed the exam?
(b) What is the expected number of students who passed the exam?
(c) How many students must take the exam to make the probability at least
0.99 that a student will pass the exam?
Notes:
1) will have to update this after work
2) will have to update this after work
3) part b), E[X(X-1)]
4) part b), finding E[X^2]
5) Just part c) here
1) An urn contains 10 marbles in which 3 are black. Four of the marbles are
selected at random (without replacement) and are tested for the black color.
Define the random variable X to be the number of the selected marbles that
are not black.
(a) Find the probability mass function of X.
(b) What is the cumulative distribution function of X?
(c) Find the expected value of X.
2) A box contains 7 marbles of which 3 are red and 4 are blue. Randomly select
two marbles without replacement. If the marbles are of the same color then
you win $2, otherwise you lose $1. Let X be the random variable representing
your net winnings.
(a) Find the probability mass function of X.
(b) Compute E(2X)
3) Suppose that X is a discrete random variable with probability mass function
p(x) = cx2
, x = 1, 2, 3, 4.
(a) Find the value of c.
(b) Find E(X) and E(X(X − 1)).
(c) Find Var(X).
4) A box contains 3 red and 4 blue marbles. Two marbles are randomly selected
without replacement. If they are the same color then you win $2. If they are
of different colors then you lose $ 1. Let X denote the amount you win.
(a) Find the probability mass function of X.
(b) Compute E(X) and E(X^2).
(c) Find Var(X).
5) The probability of a student passing an exam is 0.2. Ten students took the
exam.
(a) What is the probability that at least two students passed the exam?
(b) What is the expected number of students who passed the exam?
(c) How many students must take the exam to make the probability at least
0.99 that a student will pass the exam?
Notes:
1) will have to update this after work
2) will have to update this after work
3) part b), E[X(X-1)]
4) part b), finding E[X^2]
5) Just part c) here
Permutations and combinations questions
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