Binomial random variables
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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.

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Binomial Random Variables

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Binomial random variables

Binomial Random Variables


Binomial experiment

Binomial experiment

  • A sequence ofn trials(called Bernoulli trials), each of which results in either a “success” or a “failure”.

  • The trials are independent and so theprobability of success, p, remains the same for each trial.

  • Define a random variable Y as the number of successes observed during the n trials.

  • What is the probability p(y), for y = 0, 1, …, n ?

  • How many successes may we expect? E(Y) = ?


Returning students

Returning Students

  • Suppose the retention rate for a school indicates the probability a freshman returns for their sophmore year is 0.65. Among 12 randomly selected freshman, what is the probability 8 of them return to school next year?

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.


12 trials 8 successes

12 trials, 8 successes

  • To find the probability of this event, consider the probability for just one sample point in the event.

  • For example, the probability the first 8 students return and the last 4 don’t.

  • Since independent, we just multiply the probabilities:


12 trials 8 successes1

12 trials, 8 successes

  • For the probability of this event, we sum the probabilities for each sample point in the event.

  • How many sample points are in this event?

  • How many ways can 8 successes and 4 failures occur?

  • Each of these sample points has the same probability.

  • Hence, summing these probabilities yields


Binomial probability function

Binomial Probability Function

  • A random variable has a binomial distribution with parameters n and p if its probability function is given by


Binomial random variables

Rats!

  • In a research study, rats are injected with a drug. The probability that a rat will die from the drug before the experiment is over is 0.16. Ten rats are injected with the drug.

What is the probability that at least 8 will survive?

Would you be surprised if at least 5 died during the experiment?


Quality control

Quality Control

  • For parts machined by a particular lathe, on average, 95% of the parts are within the acceptable tolerance.

  • If 20 parts are checked, what is the probability that at least 18 are acceptable?

  • If 20 parts are checked, what is the probability that at most 18 are acceptable?


Binomial theorem

Binomial Theorem

  • As we saw in our Discrete class, the Binomial Theorem allows us to expand

  • As a result, summing the binomial probabilities, where q = 1- p is the probability of a failure,


Mean and variance

Mean and Variance

  • If Y is a binomial random variable with parameters n and p, the expected value and variance for Y are given by


Deriving expected value

When y = 0, the summand is zero. Just as well start at y = 1.

Deriving Expected Value


And deriving

=1

And deriving…


Derivingvariance

DerivingVariance?

Just the highlights (see page 104 for details).

“fairly common trick” to use E[Y(Y-1)] to find E(Y2)


Binomial random variables

Rats!

  • In a research study, rats are injected with a drug. The probability that a rat will die from the drug before the experiment is over is 0.16. Ten rats are injected with the drug.

  • How many of the rats are expected to survive?

  • Find the variance for the number of survivors.


Geometric random variables

Geometric Random Variables


Your 1 st success

Your 1st Success

  • Similar to the binomial experiment, we consider:

  • A sequence ofindependent Bernoulli trials.

  • The probability of “success” equals p on each trial.

  • Define a random variable Y as the number of the trial on which the 1st success occurs. (Stop the trials after the first success occurs.)

  • What is the probability p(y), for y = 1,2, … ?

  • On which trial is the first success expected?


Finding the probability

(S)

S

(F, S)

S

F

(F, F, S)

S

F

(F, F, F, S)

S

F

….

Finding the probability

  • Consider the values of Y:y = 1: (S)y = 2: (F, S)y = 3: (F, F, S)y = 4: (F, F, F, S)and so on…

p(1) = pp(2) = (q)( p)p(3) = (q2)( p)p(4) = (q3)( p)


Geometric probability function

Geometric Probability Function

  • A random variable has a geometric distribution with parameter p if its probability function is given by


Success

(D)

D

(G, D)

D

G

(G, G, D)

D

G

G

Success?

  • Of course, you need to be clear on what you consider a “success”.

  • For example, the 1st success might mean finding the 1st defective item!


Geometric mean variance

Geometric Mean, Variance

  • If Y is a geometric random variable with parameter p the expected value and variance for Y are given by


Deriving the mean

Deriving the Mean


Deriving variance

Deriving Variance

Using the “trick” of finding E[Y(Y-1)] to get E(Y2)…


Deriving variance1

Deriving Variance

Now, forming the second moment, E(Y2)…

And so, we find the variance…


At least a trials 3 55

At least ‘a’ trials? (#3.55)

  • For a geometric random variable and a > 0,show P(Y > a) = qa

  • Consider P(Y > a) = 1 – P(Y < a)

    = 1 – p(1 + q + q2 + …+ qa-1)

    = qa , based on the sum of a geometric series


At least b more trials

At least b more trials?

  • Based on the result, it follows P(Y > a + b) = qa+b

  • Also, the conditional probabilityP(Y > a + b | Y > a ) = qb = P(Y > b)

    “the memoryless property”


No memory

No Memory?

  • For the geometric distribution P(Y > a + b | Y > a ) = qb = P(Y > b)

  • This implies P(Y > 7 | Y > 2 ) = q5 = P(Y > 5)“knowing the first two trials were failures, the probability a success won’t occur on the next 5 trials”

    as compared to “just starting the trials and a success won’t occur on the first 5 trials” same probability?!


Estimating p example 3 13

Estimating p (example 3.13)

  • Considering implementing a new policy in a large company, so we ask employees whether or not they favor the new policy.

  • Suppose the first four reject the new policy, but the 5th individual is in favor of the policy.

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?


What value of p is most likely

What value of p is most likely?

  • We wish to find the value of p which would make it highly probable that the 5th individual turns out to be first “success”.

  • That is, let’s maximize the probability of finding the first success on trial 5, where p(5) = (1- p)4 p

  • For what value of p is this probability a max?


Find the extrema

Find the Extrema

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”


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