Chapter 4 Discrete Random Variables Flashcards
Variable (Random Variable)
Variable (Random Variable): A characteristic of interest in a population being studied. Common notation for variables are upper case letters. In other words, a random variable is a set of possible values from a random experiment.
X,Y,Z,…; common notation for a specific value from the domain (set of all possible values of a variable) are lower case lettersx,y,z,….
For example, let X = the number of heads you get when you toss three fair coins. The sample space for the toss of three fair coins is TTT; THH; HTH; HHT; HTT; THT; TTH; HHH. Then, x= 0, 1, 2, 3.
Note:*It’s common X is in words and x is a number
Probability Distribution
Probability Distribution – a probability distribution is a graph, table, or function that gives the probability for each value of the random variable.
Probability Distribution Function (PDF)
Probability Distribution Function (PDF): A mathematical description of a discrete random variable (RV), given either in the form of an equation (formula) , or in the form of a table listing all the possible outcomes of an experiment and the probability associated with each outcome. In other words, is some function that may be used to define a particular probability distribution.
A Discrete probability distribution function has two characteristics.What are they?
A Discrete probability distribution function has two characteristics:
·
Each probability is between 0 and 1, inclusive.
o 0 ≤ f(x) ≤ 1
· The sum of the probabilities is 1.
o ∑ f(x) = 1
How to calculate the Mean of a Discrete Random Variable?
How to calculate Standard Deviation of a Discrete Random Variable?
Binomial Distribution
Binomial Distribution - A distribution involving things with only 2 possible outcomes, such as the tossing of a coin.
What are the conditions that satisfies a binomial experiment?
1. The experiment is repeated for a fixed number of trials, where each trial is independent of the other trials.
2. There are only two possible outcomes of interest for each trial. The outcomes can be classified as a success (S) or as a failure (F).
3. The probability of a success, P(S), is the same for each trial.
4. The random variable, x, counts the number of successful trials.
or
1.) Fixed number of trails
2.) Trails must be independent of the other trails.
3.) Each trail outcome fits in the categories “Success” (S) & “Failure”(F)
4.) Probability P(S) must remain constant for each trail
Binomial Distribution Function
How to find Mean, Standard Deviation, and Variance for Binomial Distribution Function?
Recall about standard deviation:
- ) 68% values fall within ±1σ of the mean
- ) 95% values fall within ±2σ of mean…Max usual = µ +2σ; Min usual = µ -2σ
- ) 99.7 values fall within ±3σ of mean
The cumulative distribution function (cdf)
The cumulative distribution function (cdf) is the probability that the variable takes a value less than or equal to x. That is P[X≤x]
Poisson distribution
Poisson distribution - The Poisson distribution is a discrete probability distribution for the counts of events that occur randomly in a given interval of time (or space).In other words a Poisson distribution is a statistical distribution that shows how many times an event is likely to occur within a specified period of time.
Example
The number of patients arriving in an emergency room between 10 and 11 pm
Number of typing errors on a page
The number of births per hour during a given day
Assumptions
- x is the number of times an event occurs in an interval and x can take values 0, 1, 2, ….
- The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.
- The rate at which events occur is constant.
- Two events cannot occur at exactly the same instant
Poisson Distribution Function
·
The Poisson Distribution Function can approximate the Binomial Distribution if the following what is true?
1.) n > 100
2.) n * p < 10
n = number of trails
p = probability of success in one trail
