Chapter 2 Background Flashcards
Basics of Probability and Statistics
probability
Probability is the likelihood that some events will happen in some probability space
the study of quantifying the likelihood of particular events occurring.
The probability of a particular outcome of an experiment is the ratio of the number of ways that outcome can occur to all possible outcomes of the experiment
all the outcomes are equally likely to occur, the probability is…
If all the outcomes are
| equally likely, as in the case of a fair die, then the probability of each is 1/n.
if A and B are independent, the probability of them both occur is ……
If A and B are two independent events then the probability of them both occurring is the product of the probabilities. P(A&B) = P(A) * P(B)
Suppose you rolled the fair die twice. What is the probability of rolling the same number two times in a row?
Since we don’t care what the outcome of the first roll is, its probability is 1. The second roll of the dice has to match the outcome of the first, so that has a
| probability of 1/6. The probability of both events occurring is 1 * 1/6.
Now consider the experiment of rolling 2 dice, one red and one green. Assume the dice are fair and not loaded. How many distinct outcomes are possible?
36
Rolling these two dice, what’s the probability of rolling a 10?
3/36
What sum is the most likely when rolling these two dice?
7
The probability of at least one of two events, A and B, occurring
The probability of at least one of two events, A and B, occurring is the sum of their individual probabilities minus the probability of their intersection. P(A U
| B) = P(A) + P(B) - P(A&B).
Back to rolling two dice. Which expression represents the probability of rolling an even number or a number greater than 8?
(18+10-4)/36
It follows that if A and B are disjoint or mutually exclusive, i.e. only one of them can occur, then
then P(A U B) = P(A)+P(B)
Which of the following expressions represents the probability of rolling a number greater than 10?
(2+1)/36
If you’re dealt a hand of 5 cards, what is the probability of getting all 5 of the same value?
What is the probability of drawing a face card?
Suppose you draw a face card and don’t replace it in the deck. What is the probability that when you draw a second card it also will be a face card of the same
| suit?
0
12/52
2/51
PMF
CDF
Probability Mass Function deal with discrete random variables. P(X)=p^x*(1-p)^(1-x)
Probability Density Function deal with continuous random variables.
it “is a function that describes the relative likelihood
| for this random variable to take on a given value. The probability of the random variable falling within a particular range of values is given by … the area
| under the density function but above the horizontal axis and between the lowest and greatest values of the range.” (remember the blue triangle)
The cumulative distribution function (CDF) of a random variable X, either discrete or continuous, is the function F(x) equal to the probability that X is less than
| or equal to x
F(x)
CDF vs. PDF
survivor function S(x)
The term ‘x’ is the base, x/2 is the height
x*x/4
When the random variable is continuous, as in the example, the PDF is the derivative of the CDF
PDF (the line represented by the diagonal)
The survivor function S(x) of a random variable X is defined as the function of x equal to the probability that the random variable X is greater than the value x.
quantilte and percentile
The quantile v of a CDF is the point x_v at which the CDF has the value v. More precisely, F(x_v)=v.
A percentile is a quantile in which v is expressed as a
| percentage.
