MATH 180A Lecture 12

Example. Taco Villa makes many tacos per day, and they almost always get each customer’s taco order correct. On 75% of the days, they get every taco order correct, and on the other 25%, they get at least one taco order wrong. What is the probability that tomorrow, Taco Villa gets exactly 3 taco orders wrong?

W = # of taco orders get wrong tomorrow.
Not .25 chance they get each order wrong. We don’t n or p, but we know n is large, p is small.

Continuous random variable
Discrete

Section 3.3

Expectation
Discrete Random Variables

Suppose that a discrete distribution on has pmf . Hang a weight of mass at each point . Where is the center of mass? In other words, where should fulcrum be placed to balance the weights? For example, take

Expectation/Mean/Average

Suppose is a discrete random variable w/ pmf . Its expectation/mean/average is the real number

Example. Let be the outcome of a fair die roll. Calculate .

expectation doesn’t have to be a possible value

Examples.

1. Suppose . Calculate .
   
2. Suppose . Calculate .
   to be continued…
3. Suppose . Calculate .
   

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take derivative wrt p



4. Suppose . Calculate .

reindex

Theorem: Expectation is linear: If $X,Y$ are random variables defined on the same probability space and $c\in \mathbb{R}$, then

and

Proof.

Example. We return to . Calculate .