There are n prizes, with values $1, $2,...,$n. You get to choose k random prizes, without replacement. What is the expected total value of the prizes you get? What is the variance?
There are n prizes, with values $1, $2,...,$n. You get to choose k random prizes, without...
Therom 1.8.2
n choose k = (n choose n-k)
n choose k = (n-1 choose K) + (n-1 choose K-1)
2n = summation of (n choose i )
please use the induction method
(a) (10 pts) Show that the following equality holds: n +1 + 2 Hint: If you proceed by induction, you might want to use Theorem 1.8.2. If you search for a combinatorial proof, consider the set X - (i,j, k): 0 S i,j< k< n) (b) (10...
Suppose that n students are selected at random without replacement from a class containing 28 students, of whom 8 are boys and 20 are girls. We assume that 0 < n < 28. Let X denote the number of boys that are obtained. Answer the following questions: a (4 marks) State the distribution of X, with parameters b (1 mark) Write down the possible values of X c (1 mark) Express E(X) in terms of n. d (4 marks) For...
5. Three boxes are numbered 1, 2 and 3. For k 1, 2, 3, box k contains k blue marbles and 5 - k red marbles. In a two-step experiment, a box is selected and 2 marbles are drawn from it without replacement. If the probability of selecting box k is proportional to k, then the probability that two marbles drawn have different colours is 6. Two balls are.dropped in such a way that each ball is equally likely to...
Suppose that Y is a random variable with the probability mass function, 2 k PſY = k] = nom 1, for k=0, 1, ..., n - 1, n (n − 1)? where n > 2. 1. Derive the expected value of Y. 2. Evaluate the second moment of Y. 3. Determine the variance of Y.
Write a c++ program where it computes n choose k by using the formula: n(n-1)(n-2)...(n-k+1)/k!. But do not use algorithms where you can use only use integer arithmetic. How is this better than writing a program that uses the formula n!/k!(n-k)!
The random variable X takes the values -2, -1 and 3 according to the following probability distribution: -2 3k -1 2k 3 3k px(x) i. Explain why k = 0.125 and write down the probability distribution of X. ii. Find E(X), the expected value of X. iii. Find Var(X), the variance of X.
Problem 3 A discrete random variable Y takes values {k= 0, 1, 2, ...,} such that PLY Z k} = ()* for k 20. 1. Derive P[Y = k) for any k > 0. 2. Evaluate expectation, E[Y] = 3. Given E[Y(Y - 1)] = 15 , find variance of Y, Var[Y] =
Let X be a discrete random variable with values in N = {1, 2,...}. Prove that X is geometric with parameter p = P(X = 1) if and only if the memoryless property P(X = n + m | X > n) = P(X = m) holds. To show that the memoryless property implies that X is geometric, you need to prove that the p.m.f. of X has to be P(X = k) = p(1 - p)^(k-1). For this, use...
) 11. Suppose you have 10 gift bags and 2 contain prizes worth $50. The other gift bags have items worth $20. (3) a. Find the probability you select two bags with one $50 prize and one $20 prize. (3) b. Find the probability that you select two bags and both contain $50 prizes. (3) c. Find the probability that you select three bags and two contain $50 prizes and one contains a $20 prize. (3) d. What is the...
Using python, write a program to get the sample mean values with n=5 samples for 1000 experiments. Each sample has a normal distribution of ~N(0,1). This generates 1000 sample mean values. Consider the sample mean value to be a random variable. Find its mean and variance. (this should match part a)