Random Variables Chapter 4

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1

What are random variables?

-Variables that assumes numerical values associated with the outcomes of a random experiment, where one ( and only one) numerical value is assigned to each elementary outcome. So, we think of random variables as functions whose domain is the sample space of a random experiment and whose range is the set of possible values of the variable.

2

What are the two major types of random variables?

-Discrete

-Continuous

3

What is a Probability distribution

-A probability distribution is a function that maps each possible value a random variable can take to the probability that it takes that value. Because the form and interpretation of probability distributions are quite different for discrete and continuous random variables, however we distinguish the two types of distributions with slightly different terms that have crucially different interpretations.

4

What is Discrete random variables?

-The distribution of a discrete random variable is generally expressed via a probability mass function that maps each possible value in the domain set of the random variable to a unique mass of probability associated with that value. Rather than algebraic equations, we begin by summarizing the mapping from variable values to probabilities with a simple distribution table. Because discrete values can be ordered and listed, we can tabulate the possible values of a discrete random variable together with the probabilities associated with those values.

5

What is an example of Discrete random variables?

-One of the simplest of random experiments is tossing a coin

6

What are basic measure of location and spread descriptions?

-Mean and Standard deviation

7

What is E(X)?

-Expected value of X

8

What are the Discrete Random Variables we learned?

-Binomial Random Variables

-Hypergeometric Random variables

-Geometric Random variables

-Poisson Random Variables

9

What are the defining characteristics of Binomial Random Variables?

1. a simple experiment with only 2 possible outcomes, one of which we denote the outcome of interest;

2. the simple experiment is repeated a fixed number n times;

3. successive repetitions of the experiment (called trials) are independent;

4. the probability π of the outcome of interest stays fixed throughout all trials (implying that the probability of the complementary outcome stays fixed at 1 − π);

5. our variable of interest is the count of the total number of times the outcome of interest occurs over the n trials.

10

What are the defining characteristics of Hypergeometric Random Variables?

1. we sample n elements at random and without replacement from a population of N objects of which r have a certain trait of interest and the other N − r do not;

2. we define the random variable X to be the number of the n elements sampled that have the trait of interest.

11

What are the defining characteristics of Geometric characteristics?

1. a random experiment with only two possible outcomes;

2. repeated until one particular outcome of interest occurs;

3. where successive repetitions of the experiment (called trials) are independent;

4. probabilities of both of the two possible outcomes remain fixed across all trials;

5. our variable is the count of the total number of trials performed in order to obtain the first outcome of interest

12

What are the defining characteristics of Poisson Random Variables?

1. The experiment consists of counting the number of times a certain event occurs during some fixed unit of measurement, such as a period of time, a length, an area, etc.

2. The probability an event occurs in any one unit of measurement is constant.

3. The numbers of events to occur in any two disjoint intervals of measurement are independent.

4. The expected number of events to occur in any particular interval is denoted λ (lowercase Greek letter lambda).