I need assistance to solve the following.
Question:
Andrea Anti-Ponzi is a client who wishes to use a hash table that you design. You, the computer scientist, have decided to use a hash table with m slots that resolves collisions using chaining. You make the assumption of simple uniform hashing. Andrea, being mostly familiar with ponzi schemes, strongly dislikes chains of length two or greater. Therefore, you need to compute the expected number of hash slots that have chains of length two or greater. For parts (a)-(e) below, suppose that we have inserted n distinct keys into the hash table, and let j ? {0, 1, . . . , m ? 1} be arbitrary.?
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(a) What is the probability that none of the keys hash to slot j??
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(b) What is the probability that exactly one key hashes to slot j??
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(c) What is the probability that at least two keys hash to slot j??
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(d) What is the expected number of hash slots which store two or more elements (i.e., hash slots whose chains are of length two or more)??
2. Let X1, X2, ..., Xn be n identically independent distributed Bernoulli random variables with parameter pi = p, i = 1, 2, ..., n. We learned in class that Y = X1 + X2+...+X, has Binomial distribution. 2a. Identify the parameters of this Binomial distribution. 2b. Find the expectation of the Binomial distribution. 2c. Find the variance of this Binomial distribution.In each situation below, is it reasonable to use a binomial distribution for the random variable X? Give reasons for your answerin each case. {a} A random sample of students in a fitness study. X is the mean systolic blood pressure of the sample. tries, a binomial distnbution is reasonable. X can only take on two values No, a binomial distnbution is not reasonable. X is not a count of successes. No; a binomial tstribution is not reasonable. Binomial distnbutions cannot be used with random samples No, a binomial tstribution is not reasonable. it should be associated with a population \"ms a binomial tstribution is reasonable. X is a mean of the binomial distribution. [bi A manufacturer of running shoes picks a random sample o'fthe production of shoes each day for a detailed inspection. Today's sample of 20 pairs of shoes indudes 1 pairwith a defect. No; a binomial bstribution is not reasonable. Dne defect in a sample of 2B is not a large enough percentage. No, a binomial distribution is not reasonable. Binomial distributions cannot be used with random samples. \"ms a binomial distribution is reasonable. p is the probability of a defective pair. "ms a binomial distribution is reasonable. p is the number of defective shoes from today's sample. No, a binomial distribution is not reasonable. Cine defect is not a large enough count. {c} A nutrition study d'iooses an SR5 of college students Tl'iey are asked whether or not they usually eat at least ve servings of fruits or vegetables per day. X is the nurrberwho saythat they do. \"lies; a binomial distribution is reasonable. n is the number of students chosen from the sample and X is the number of servings of fmits and vegetables they eat. No, a binomial tstribuijon is not reasonable. A student might eat less than ve senaings offnJits and vegetables but could daim othenlvise. Yes, a binomial distnbution is reasonable. p is the percentage of students d'iosen from the population and n is the number of servings of fmits. and vegetables they eat. Yes, a binomial distnbution is reasonable. n is the number of students in the sample and p is the probabilitythat a student eats at least five servings o'ffruits and vegetables No, a binomial tstribution is not reasonable. Binomial distnbutions cannot be used with random samples 2. Let X1, X2, ..., Xn be n identically independent distributed Bernoulli random variables with parameter p; = p, i = 1, 2, ..., n. We learned in class that Y = X1 + X2+. . . + X, has Binomial distribution. 2a. Identify the parameters of this Binomial distribution. 2b. Find the expectation of the Binomial distribution. 2c. Find the variance of this Binomial distribution
