1. Random variables measure an unpredictable quantity in a certain experiment. A

Question

1. Random variables measure an unpredictable quantity in a certain experiment. A random variable is a function that translates each outcome in a particular sample space into a real number( which is easy to deal with). They don't have a value by themselves, but can take on different values in a trial of the experiment. Remember that randomness comes from the experiment. Once we perform the experiment we can observe what happened. Note: A degenerate distribution comes from a random variable that always produces the same real number. Random variable X: A fair coin is flipped once. X(H) = 1, X(T) = -1. Random variable Y: A fair die is tossed once. X(1) = 1, X(2) = 1, X(3) = 1, X(4) = -1, X(5) = -1, X(6) = -1.

a) Does random variable X have the same distribution as random variable Y.

b) Are X and Y the same random variable?

2 The hypergeometric is sampling “without replacement.” Imagine you have this bag of marbles with 35 marbles and 22 of them are black. We will define a “success” as drawing a black marble. You will be drawing 15 marbles as a sample.

(a) Let’s say you draw one marble. Call this r.v. X. Is it hypergeometric? If not, what distribution is this? (b) The hypergeometric distribution has three parameters. What are the parameters specifically for X? (c) Write the PMF, p(x) for the r.v. X where x is the number of successes. (d) What is the support of this r.v.?

3. Two 6 sided dice are rolled. The first die is Red with sides 1,3,4,4,5,6 and the second die is Green with sides 2,2,3,3,5,6. Random variable X = max( 2 rolls). Each side of both dice has a 1/6 probability. a) Find the distribution of the pmf for X = b) Find the distribution for F(X) = c) Find the Prob( 2 < x < 5) = d) Find the Prob( x > 4.1) = e) Find the prob(3 <x <=6)

4 .What is the minimum number of times you should toss a fair coin if you want a probability of at least .97 of getting heads at least once?

a)  P(x) = x/c where c is a constant. The support of X = 1,2,3,,,,n where X is a discrete( finite) R.V.   Find the value of c that makes this a valid pmf.

b) Lets say P(x) = c where X~ discrete uniform on the integers 6,7,8,9,,,,,,200.. Find the value of c that makes this a valid pmf..

5. A restaurant can accommodate 50 people. The restaurant takes reservations for 52 people for Sunday dinner. The probability any given person making a reservation does not show up is 10 %. What is the probability that all the people who show up for Sunday dinner can be accommodated?

Explanation / Answer

5)

Probability = 52C50 * 0.90^50 * 0.10^52-50 = 0.06833

for the other question please post it individually in other question

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