5/1/17 Lecture notes (12) Handout
Tuesday, May 2, 2017 1:44 PM
Handout, Day 1:
Continuous random variables
Note: the book doesn’t cover this material well, so a supplement
on Continuous Random Variables can be found on Triton Ed.
Outcome Probability of the event
-$4 1/2
$72 (1/2)(1/6)= 1/12
-$12 (1/2)(5/6)= 5/12
X = number of trials needed to get first success Probability
1 P(1) = p
2 P(2) = qp
3 P(3) = qqp
…
X P(x) = q p
A discrete random variable takes on finitely many values or
infinitely many values which are discrete (spaces in between)
A continuous random variable is a random quantity that can
take on any value on a continuous scale (a smoothinterval of
possibilities). Examples: the amount of water you drink in a day,
how long you wait for a bus, how far you live from the nearest
grocery store.
Some awkward questions?
- How do I make a probability table for such a situation?
- What is the probability of drinking exactly 2.4314gallons
of water in a given day?
- Is the deominatorof P(A) =
equal to infinity?
For continuous random variables, we have to change how we
present the probability model (no more tables) alter how we
seek questions, generalize our definition of probability and
rethink what a probability of 0 means
From discrete to continuous random variables
Suppose we flip a coin 16 times and record how many heads we
get. That is let X = Binom (n=16, p=0.5)
When we visualize the probability table:
An outcomeis more likely if there is more area in the rectangle
for that value
We also know that sum of the areas of the bars must be 1 We also know that sum of the areas of the bars must be 1
Finally, we see that the bar heights must be at least 0 ( no
negative heights)
If we momentarilyuse dots, instead of rectangles, the picture suggests
how we could generalize just draw curves.
When we do, the meaning of the y - axis changes
With discrete random variables, two ideas are linked with probability:
height on the y-axis and area ( of a rectangle)
With continuous random variables, only area is linked to probability
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