\(0<P(x_i)<1\)
\(\sum_i P(x_i) = 1\)
| Random Process | Random Variable |
|---|---|
| Rolling a two six sided die | X = Sum of faces |
| Flip a coin 10 times | X = # Number of Tails |
| Body Dimensions | X = Foot size Y = Neck Circumference Z = Height |
Are like functions that map onto the real line
Often given the names X, Y, and Z
Capital X is the name of the variable
Lowercase x is the value the variable takes
Type of Random variables:
Continuous Random Variables
Discrete Random Variables
Get a Probability Mass Function
States their probability of any value of x
Get a Probability Density Function
This states there probability for any value less than or grater than little x.
Expectation is another word for average.
It is \(E[X] = \sum_i P(x_i) x_i\)
Let’s make a probability model for X = The value on the face of a dice.
Let’s graph the model.
Then let’s calculate its expectation.
Write the random variable.
Find the expectation for the sum of two die.
Write the probability model (as a table or graph).
Here is the sample space for rolling two die 1
Problem 3.30
General Multiplication Rule:
\(P(A \text{ and } B) = P(A|B)P(B)\)
Law of Total Probability:
\(P(A) = P(A|B_1)P(B_1)+P(A|B_2)P(B_2)+…+P(A|B_k)P(B_k)\)
Where each \(B_i\) is disjoint.
Conditional Probability
\(P(A|B) = \frac{P(B|A)P(A)}{P(B)}\)
3.14, 3.15
