00:01
For this problem to begin, we know that x is a negative binomial random variable, with parameters p equal to 0 .3, and r equal to 4.
00:16
In part a, we're asked for the expected value of x.
00:20
Now, for any negative binomial random variable, the expected value is just r times 1 minus p divided by p.
00:29
So, that will be 4 times 0 .7, divided by 0 .3, for a result of 9 .333, roughly.
00:49
For part b, we're asked for probability of x equals 20.
00:53
Now, i'll note that generally, since x is a negative binomial, the probability that it's equal to some value k, is given by k, oops, k minus r, choose r minus 1 times p to the power of r times 1 minus p to the power of k minus r.
01:19
So that means that in our particular case here the probability of x equal to k is going to be equal to k minus 4 choose 4 minus 1 so choose 3 times 0 .3 to the power of 4 times 0 .7 to the power of k minus 4.
01:42
So in part b, when we're looking for probability of x equal to 20, we'd have 20 minus 4 choose 3.
01:50
So that's going to be 16 to choose 3, which is 16 factorial divided by 3 factorial times 16 minus 3, which would be 13 factorial, times 0 .3 to the power of 4 times 0 .7 to the power of 20 minus 4...