Examples

Gambling game - tokens in bins

Consider a game like this: a coin is flipped; if then draw a token from Bin 1, if then from Bin 2.

• Bin 1 contents: 1 token $1,000, and 9 tokens $1
• Bin 2 contents: 5 tokens $50, and 5 tokens $1

It costs $50 to enter the game. Should you play it? (A lot of times?) How much would you pay to play?

Solution

(1) Setup:

Let be a random variable measuring your winnings in the game.

The possible values of are 1, 50, and 1000.

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(2) Find the PDF of :

For have

For have

For have

These add to 1, and for all other .

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(3) Find using the discrete formula:

Since , if you play it a lot at $50 you will generally make money.

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Challenge Q: If you start with $200 and keep playing to infinity, how likely is it that you go broke?

Expected value: rolling dice

Let be a random variable counting the number of dots given by rolling a single die.

Then:

Let be an RV that counts the dots on a roll of two dice.

The PMF of :

Then:

Notice that .

In general, .

Let be a green die and a red die.

From the earlier calculation, and .

Since , we derive by simple addition!

Expected value by finding new PMF

Let have distribution given by this PMF:

Find .

Solution

(1) Compute the PMF of :

PMF arranged by possible value:

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(2) Calculate the expectation:

Using formula for discrete PMF:

Variance for composite using PMF and simpler formula

Suppose has this PMF:

	1	2	3

Find using the formula with .

(Hint: you should find and along the way.)