225-Notes04 - Stat 225 Lecture Notes “Discrete Random Variables” Ryan Martin Spring 2008 1 Random Variables For many experiments we are more

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Unformatted text preview: Stat 225 Lecture Notes “Discrete Random Variables” Ryan Martin Spring 2008 1 Random Variables For many experiments, we are more interested in a characteristic of the outcome rather than the outcome itself. For example, you are more interested how many lottery numbers you have correct than what those numbers are. The notion of a variable is introduced in elementary algebra courses. Such a concept extends to the situation in which the values taken by a variable are determined by a random mechanism or experiment. • In algebra and calculus, a variable, often denoted by x , is used to represent an arbitrary number, vector, etc. The reason for this is that the properties derived in those courses DO NOT depend on the actual value of x . • In probability, a random variable, often denoted by X , is used to represent the outcome of a random experiment. The probabilistic properties we study DO NOT depend on the particular outcome, but only on the experiment. Consider two simple experiments which we will use for illustration. I. Randomly choose a person in the room and count how many siblings they have. II. Randomly choose a battery from a production line and measure its lifetime. The common feature of these examples is that the quantity of interest (number of sib- lings, battery lifetime) is simply a “function” of, or measurement taken on, the observed outcome ω ∈ Ω. Definition 1.1. A random variable is a real-valued function whose domain is the sample space of a random experiment; i.e. X : Ω → R . That is, for any “simple” outcome ω ∈ Ω, we can take a real-valued measurement X ( ω ) on this outcome. In Experiment I above, if X ( ω ) is the number of siblings person ω of this class has, then X (Ryan) = 2 since I have two sisters. Example 1.2. Let Ω be the sample space of outcomes when a fair die is rolled twice and let X be the sum of the two resulting faces. Then X (1 , 2) = 3, X (2 , 4) = 6, etc. 1 We are interested in the probability of events that are defined by random variables; recall questions like “What is the probabiltity that the sum of two rolls of a fair die is even?” In general, if we let A = { ω ∈ Ω : X ( ω ) ∈ K } for some K ⊂ R , then P ( A ) = P ( { ω ∈ Ω : X ( ω ) ∈ K } ) = P ( X ∈ K ) , where the last term is just shorthand notation for the middle term. Example 1.3. If Ω consists of the outcomes of two rolls of a die and X is the sum, then the event that the sum is even can be written as A = { ω ∈ Ω : X ( ω ) ∈ { 2 , 4 , 6 , 8 , 10 , 12 }} and the probability is P ( A ) = P (sum is even) = P ( X ∈ { 2 , 4 , 6 , 8 , 10 , 12 } ) = 1 2 . Example 1.4. Probabilists and statisticians often want to conduct experiments to study the performance of their methods, and computers are used to make this experimentation time and cost efficient (we can tell a computer to “flip a coin 10,000 times” and it will be done in an instant). However, the computer is not actually flipping coins! What thebe done in an instant)....
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This note was uploaded on 06/15/2008 for the course STAT 225 taught by Professor Martin during the Spring '08 term at Purdue University-West Lafayette.

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225-Notes04 - Stat 225 Lecture Notes “Discrete Random Variables” Ryan Martin Spring 2008 1 Random Variables For many experiments we are more

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