06-Random Variables complete

Sections 85 and 86 look at properties for continuous

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Unformatted text preview: te random variables called the Binomial. The Binomial random variable arises in situations where you are counting the number of successes that occur in a sample. Sections 8.5 and 8.6 look at properties for continuous random variables and then spend more time studying the family of uniform random variables and normal random variables. In later chapters you will be introduced to more models for continuous random variables that are primarily used in statistical testing problems. Below is a summary of the types of random variables we will work with in this course. Random Variable Continuous Discrete Binomial Uniform Normal More to come Technical Note: Sometimes a random variable fits the technical definition of a discrete random variable but it is more convenient to treat it, that is, model it, as if it were continuous. In section 8.7 we will see when it is reasonable to model a discrete binomial random variable as being approximately normal. Finally Section 8.8 will show us how to model sums and differences of random variables. Some general notes about random variables are: random variables will be denoted by capital letters, e.g. X, Y, Z; outcomes of random variables are represented with small letters, e.g. x, y, z. So when we express probabilities about the possible value of a random variable we use the capital letter. For example, the probability that a random variable takes on the value of 2 would be expressed as P(X = 2). ). P(X=x) => so P(X = 2) is ok but not P(x = 2) 42 8.2 General Discrete Random Variables A discrete random variable, X , is a random variable with a finite or countable number of possible outcomes. The probability notation your text uses for a Discrete Random Variable is given next: Discrete Random Variable: X = the random variable. k = a number that the discrete random variable could assume. P(X = k) is the probability that the random variable X equals k. The probability distribution function (pdf) for a discrete random variable X is a table or rule that assigns probabilities to the possible values of the X. One way to show the istribution is through a table that lists the possible values and their corresponding probabilities: Value of X x1 x2 x3 … Probability p1 p2 p3 … Two conditions that must always apply to the probabilities for a discrete random variable are: Condition 1: The sum of all of the individual probabilities must equal 1. Condition 2: The individual probabilities must be between 0 and 1. A probability histogram or better yet, a probability stick graph, can be used to display the distribution for a discrete random variable. We will make one soon! The x‐axis represents the values or outcomes. The y‐axis represents the probabilities of the values or outcomes. The cumulative distribution function (cdf) for a discrete random variable X is a table or rule that provides the probabilities P(X k) for any real number k. Generally, the term cumulative probability refers to the probability that X is less than or equal to a particular value. Try It! Psychology Experiment A psychology experiment on the behavior of young children involves placing a child in a designated...
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