unit 1 - chapter 4 - discrete distribution Flashcards
properties of the binomial distribution - the bernoulli process
- Bernoulli Process (2 outcomes) binomial distribution
P = success
Q = failure
Trials are independent
Trials are finite
Proactively and/or retroactively (why did that happen)
properties of the geometric distribution - the bernoulli process
- Bernoulli Process (2 outcomes) binomial distribution
P = success???
Q = failure????
Independent trials
Trials stop at first success
Theoretically the genetic is an infinite distribution
Infinite distribution
Properties of hypergeometric distribution
Finite population
Group of interest
Remaining group
Sampling without replacement
Depend trials
Probability of success changes across trials
binomial
variable of interest -
number of trial -
fixed -
distribution defined by -
Variable of interest - number of success trial of interest
Number of trial - Limited
Fixed - Number of trials
Distribution defined by - N and p
geometric
variable of interest -
number of trial -
fixed -
distribution defined by -
Variable of interest - Number of trials (number of failures) until first success
Number of trial - Indefinite
Fixed - Number of success
Distribution defined by - p
law of large numbers
- As the number of trials increase the actual value moves toward the expected value
- As the sample size increases the observed statistic moves toward the true parameter
Mean likely to happen
Casino games: dice games
We expect the middle number to happen (2-12 so 7)
Setting rates for insurance
Sports performance
Markets trading
Setting the with w=vskye of expected outcomes
Sampling
Growth cycles of companies
for the hypergeometric to work….
- the population must be dividable into two and only two independent subsets (aces and non-aces in our example). The random variable X = the number of items from the group of interest.
- the experiment must have changing probabilities of success with each experiment (the fact that cards are not replaced after the draw in our example makes this true in this case). Another way to say this is that you sample without replacement and therefore each pick is not independent.
- the random variable must be discrete, rather than continuous.
binomial distribution
q = (1-p)
A more valuable probability density function with many applications is the binomial distribution.
This distribution will compute probabilities for any binomial process. A binomial process, often called a Bernoulli process after the first person to fully develop its properties, is any case where there are only two possible outcomes in any one trial, called successes and failures.
It gets its name from the binary number system where all numbers are reduced to either 1’s or 0’s, which is the basis for computer technology and CD music recordings.
for the binomial formula to work…
For the binomial formula to work, the probability of a success in any one trial must be the same from trial to trial, or in other words, the outcomes of each trial must be independent.
Flipping a coin is a binomial process because the probability of getting a head in one flip does not depend upon what has happened in PREVIOUS flips.
(At this time it should be noted that using p for the parameter of the binomial distribution is a violation of the rule that population parameters are designated with Greek letters.
probability theory
In probability theory, under certain circumstances, one probability distribution can be used to approximate another. We say that one is the limiting distribution of the other.
If a small number is to be drawn from a large population, even if there is no replacement, we can still use the binomial even thought this is not a binomial process. If there is no replacement it violates the independence rule of the binomial.
when can we use a binomial even if it is a hypergeometric distribution
we can use the binomial to approximate a probability that is really a hypergeometric distribution if we are drawing fewer than 10 percent of the population, i.e. n is less than 10 percent of N in the formula for the hypergeometric function.
The rationale for this argument is that when drawing a small percentage of the population we do not alter the probability of a success from draw to draw in any meaningful way.
important: 4 characteristics of a binomial experiment
- There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter n denotes the number of trials.
- The random variable, 𝑥, number of successes, is discrete.
- There are only two possible outcomes, called “success” and “failure,” for each trial. The letter p denotes the probability of a success on any one trial, and q denotes the probability of a failure on any one trial. p + q = 1.
- The n trials are independent and are repeated using identical conditions. Think of this as drawing WITH replacement. Because the n trials are independent, the outcome of one trial does not help in predicting the outcome of another trial.
bernoulli trial
bernoulli Trial (named after Jacob Bernoulli who, in the late 1600s, studied them extensively). A binomial experiment takes place when the number of successes is counted in one or more Bernoulli Trials.
important: 4 characteristics of geometric probability density function
X ~ G(p)
The geometric probability density function builds upon what we have learned from the binomial distribution. In this case the experiment continues until either a success or a failure occurs rather than for a set number of trials.
- There are one or more Bernoulli trials with all failures except the last one, which is a success. In other words, you keep repeating what you are doing until the first success. Then you stop
- In theory, the number of trials could go on forever.
- The probability, p, of a success and the probability, q, of a failure is the same for each trial. p + q = 1 and q = 1 − p. For example, the probability of rolling a three when you throw one fair die is 1/6. This is true no matter how many times you roll the die.
- X = the number of independent trials until the first success.
