McMillan Math
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Binomial expectation and variance

(1) Suppose we have repeated Bernoulli trials with .

The sum is a binomial variable: .

(2) We know and .

The summation rule for expectation:

(3) The summation rule for variance:

Multinomial covariances

Each trial of an experiment has possible outcomes labeled with probabilities of occurrence . The experiment is run times.

Let count the number of occurrences of outcome . So .

Find .

Solution

Notice that is also a binomial variable with success probability . (‘Success’ is an outcome of either or .)

The variance of a binomial is known to be for whatever relevant and .

So we compute by solving:

Hats in the air

All sailors throws their hats in the air, and catch a random hat when they fall down.

How many sailors do you expect will catch the hat they own? What is the variance of this number?

Solution

Strangely, the answers are both 1, regardless of the number of sailors. Here is the reasoning:

(1) Let be an indicator of sailor catching their own hat. So when sailor catches their own hat, and otherwise. Thus is Bernoulli with success probability .

Then counts the total number of hats caught by original owners.

(2) Note that .

Therefore:

(3) Similarly:

We need and .

(4) Use . Observe that . Therefore:

(5) Now for covariance:

We need to compute .

Notice that when and both catch their own hats, and 0 otherwise.

We have:

Therefore:

(6) Putting everything together back in (1):

Months with a birthday

Suppose study groups of 10 are formed from a large population.

For a typical study group, how many months out of the year contain a birthday of a member of the group? (Assume the 12 months have equal duration.)

Solution

(1) Let be 1 if month contains a birthday, and 0 otherwise.

So we seek . This equals .

The answer will be because all terms are equal.

(2) For a given :

The complement event:

(3) Therefore:

Pascal expectation and variance

(1) Let .

Let be independent random variables, where:

  • counts the trials until the first success
  • counts the trials after the first success until the second success
  • counts the trials after the success until the success

Observe that .

(2) Notice that for every . Therefore:

(3) Using the summation rule, conclude: