Answer:
a) For this case we can find the cumulative distribution function first:
So then by the complement rule we have this:
![P(Y>a) = 1-F(a)= 1- [1-(1-p)^a]= 1-1 +(1-p)^a = (1-p)^a = q^a](https://tex.z-dn.net/?f=P%28Y%3Ea%29%20%3D%201-F%28a%29%3D%201-%20%5B1-%281-p%29%5Ea%5D%3D%201-1%20%2B%281-p%29%5Ea%20%3D%20%281-p%29%5Ea%20%3D%20q%5Ea)
b) 
So then we have this using independence:
We want to find the following probability:
Using the definition of conditional probability we got:
And we see that if a = 2 and b=5 we have:
c) For this case we use independent identical and with the same distribution experiments.
And the result for part b makes sense since we are interest in find the probability that the random variable of interest would be higher than an specified value given another condition with a value lower or equal.
Step-by-step explanation:
Previous concepts
The geometric distribution represents "the number of failures before you get a success in a series of Bernoulli trials. This discrete probability distribution is represented by the probability density function:"
If we define the random of variable Y we know that:
Part a
For this case we can find the cumulative distribution function first:
So then by the complement rule we have this:
Part b
For this case we can use the result from part a to conclude that:
So then we have this assuming independence:
We want to find the following probability:
Using the definition of conditional probability we got:
And we see that if a = 2 and b=5 we have:
Part c
For this case we use independent identical and with the same distribution experiments.
And the result for part b makes sense since we are interest in find the probability that the random variable of interest would be higher than an specified value given another condition with a value lower or equal.
