Binomial Random Variables. Binomial experiment. A sequence of n trials ( called Bernoulli trials ), each of which results in either a “success” or a “failure”. The trials are independent and so the probability of success, p , remains the same for each trial.
Each student either returns or doesn’t. Think of each selected student as a trial, so n = 12.
If we consider “student returns” to be a success, then p = 0.65.
What is the probability that at least 8 will survive?
Would you be surprised if at least 5 died during the experiment?
=1 1.And deriving…
Just the highlights (see page 104 for details).
“fairly common trick” to use E[Y(Y-1)] to find E(Y2)
(F, F, S)
(F, F, F, S)
….Finding the probability
p(1) = pp(2) = (q)( p)p(3) = (q2)( p)p(4) = (q3)( p)
(G, G, D)
Using the “trick” of finding E[Y(Y-1)] to get E(Y2)…
Now, forming the second moment, E(Y2)…
And so, we find the variance…
= 1 – p(1 + q + q2 + …+ qa-1)
= qa , based on the sum of a geometric series
“the memoryless property”
as compared to “just starting the trials and a success won’t occur on the first 5 trials” same probability?!
What does this tell us about the percentage of employees we might expect to favor the policy?Can we estimate the probability p of getting a favorable vote on any given trial?
Using the derivative to locate the maximum
The derivative is zero and the probability is at its maximum when p = 0.2
“the method of maximum likelihood”