The probability of success in Bernoulli is 0.7. Find the expected value and variance of the...

The probability of success in Bernoulli is 0.7. Find the expected value and variance of the number of failures until the ninth success. (The problem is to find the mean and variance of the number of failures in the negative binomial distribution given the Bernoulli probability of success.)

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Expert Solution

The Negative Binomial Distribution is a discrete probability distribution of the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of failures (denoted r) occurs. For example, if we define a 1 as failure, all non-1s as successes, and we throw a die repeatedly until 1 appears the third time (r = three failures), then the probability distribution of the number of non-1s that appeared will be a negative binomial distribution.

The probability mass function of the negative binomial distribution is

where k is the number of successes, r is the number of failures, and p is the probability of success.

Therefore here

p=0.7, r=9

When counting the number k of successes before r failures, the expected number of successes is rp/(1 − p).

Therefore mean is 9*(0.7)/(1-0.7)=21

When counting the number k of successes given the number r of failures, the variance is rp/(1 − p)².

Therefore variance is 9*0.7/(0.3)^2 = 70.

orchestra answered 1 year ago

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