Suppose you have binomial trials for which the probability of success on each tr

Question

Suppose you have binomial trials for which the probability of success on each trial is p and the probability of failure is q = 1 – p. Let k be a fixed whole number greater than or equal to 1. Let n be the number of the trial on which the kth success occurs. This means that the first k – 1 successes occur within the first n – 1 trials, while the kth success actually occurs on the nth trial. Now, if we are going to have k successes, we must have at least ktrials. So n = k, k + 1, k + 2, ... and n is a random variable. In the literature of mathematical statistics, the probability distribution for n is called the negative binomial distribution.

Negative binomial distribution
Let k 1 be a fixed whole number. The probability that the kth success occurs on trial number n is

P(n) = Cn – 1,k – 1pkqn – k

where

The expected value and standard deviation of this probability distribution are

Note: If k = 1, the negative binomial distribution is called the geometric distribution.

In eastern Colorado, there are many dry land wheat farms. The success of a spring wheat crop is dependent on sufficient moisture in March and April. Assume that the probability of a successful wheat crop in this region is about 75%. So the probability of success in a single year is p = 0.75, and the probability of failure is q = 0.25. The Wagner farm has taken out a loan and needs k = 4 successful crops to repay it. Let n be a random variable representing the year in which the fourth successful crop occurs (after the loan was made).

(a) Write out the formula for P(n) in the context of this application. (Use C(a,b) as the notation for "a choose b".)
P(n) =  

(b) Compute P(n = 4), P(n = 5), P(n = 6), and P(n = 7). (Use 4 decimal places.)

(c) What is the probability that the Wagners can repay the loan within 4 to 7 years? Hint: Compute P(4 n 7). (Use 4 decimal places.)


(d) What is the probability that the Wagners will need to farm for 8 or more years before they can repay the loan? Hint: Compute P(n 8). (Use 4 decimal places.)


(e) What are the expected value and standard deviation of the random variable n? (Use 2 decimal places.)

Interpret these values in the context of this application.

The expected year in which the fourth successful crop occurs is , with a standard deviation of .The expected year in which the third successful crop occurs is , with a standard deviation of .     The expected year in which the fifth successful crop occurs is , with a standard deviation of .The expected year in which the fourth successful crop occurs is , with a standard deviation of .

Explanation / Answer

a) p = 0.75, and the probability of failure is q = 0.25. k = 4

P(n) = (n-1)C3 * p^4 * q^(n-4) = (n-1)C3 * 0.75^4 *0.25^(n-4)

b) P(4) = 0.75^4 = 0.31640625

P(5) = 4*0.75^4*0.25 = 0.31640625

P(6) = 10* 0.75^4 * 0.25^2 = 0.19775390625

P(7) = 20* 0.75^4 * 0.25^3 = 0.098876

c) P(4 n 7) = 2*0.31640625 + 0.19775390625+0.098876

=0.929442

d)P(n>=8) = 1- P(n<=7) = 1 -0.929442= 0.070558 ,note that n can not be less than 4 , n>= k {here k =4}

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