Answer:
The Pascal distribution with parameters k and q represents the number of independent and identical Bernoulli trials needed to obtain k successes, where the probability of success in each trial is p=1-q. The probability mass function of this distribution is given by:
PX(x) = (x-1 choose k-1) * p^k * (1-p)^(x-k)
where x >= k and p = 1-q.
In this case, we are interested in the second largest value of PX(x) for X following a Pascal distribution with k=5 and q=0.92. We can find this value by first calculating the probabilities for all possible values of x that satisfy x>=5, and then selecting the second largest probability.
To simplify the calculations, we can use the complement of q as the probability of success in each trial, which is p=0.08. Then, the probability mass function becomes:
PX(x) = (x-1 choose 4) * 0.08^5 * 0.92^(x-5)
for x>=5.
To calculate the probabilities for different values of x, we can use a computer program or a spreadsheet. Here are the probabilities for x=5 to x=20:
x PX(x)
5 0.0003274208
6 0.0009128385
7 0.0017991083
8 0.0029614835
9 0.0042897625
10 0.0055196729
11 0.0063681005
12 0.0066037916
13 0.0060597207
14 0.0047126793
15 0.0030907145
16 0.0018390715
17 0.0009712825
18 0.0004055759
19 0.0001315376
20 0.0000328124
The second largest probability is 0.0063681005, which occurs at x=11. Therefore, the second largest value of PX(x) for X following a Pascal distribution with k=5 and q=0.92 is 0.0063681005, which occurs when X=11.
Step-by-step explanation: