Random variables lecture ntoes outline

Random variables lecture ntoes outline - 1 Introduction to...

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1 Introduction to random variables Example 1.1: The experiment is to toss a fair coin 2 times. Recall the sample space, S = { HH, HT, TH, TT } . which we will assume has equally-likely outcomes (i.e., each has probability 1/4). Let X = the number of “heads” we will see if we perform this experiment. X can take on values: 0, 1, or 2. For example, the event X = 1 has the outcomes { HT, TH } We can compute the probabilities of the events X = 0, X = 1, X = 2 using the equally-likely rule: P ( X = 0) = P ( { TT } ) = 1 / 4 P ( X = 1) = P ( { HT, TH } ) = 2 / 4 = 1 / 2 P ( X = 2) = P ( { HH } ) = 1 / 4 We can compute the probabilities of other events, for example: P ( X 1) = P ( { HT, TH, TT } ) = 3 / 4 We call X a random variable. 1.1 Definitions Recall the definition of an experiment from before: Definition: experiment action or process that generates outcomes. Only one outcome can occur and we are usually uncertain which outcome this will be. Definition: random variable a numerical measurement of the outcome of an experiment that has yet to be performed. We will study discrete random variables and continuous random variables. discrete: the set of possible values has a finite or countable number of elements/values. (e.g. { 0 , 1 } , {- 1 . 5 , 0 , 1 . 5 , 2 , 3 } , { 1 , 2 , 3 , . . . } continuous: the set of possible vales has an uncountably infinite number of ele- ments/values, such as intervals, (e.g. [0 , 1] , ( -∞ , )). Also, P ( X = c ) = 0 for any possible value of c . 1
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Notation: Reserve capital letters for random variables. (e.g. X, Y, Z ) Reserve lower-case letters for specific realizations of the random variables. (e.g. x, y, z ) Definition: realization of random variable the value that the random variable takes on after the experiment is performed. (e.g. if the random variable X counts the number of heads in 3 coin flips (we have yet to flip), and we carry-out this experiment and see 2 heads, then x = 2 is the realization of X . Definition: probability distribution the random variable’s possible values and the probabilities it takes on these values. Usually determined by the probability mass function for discrete random variables, and the probability density function for continuous random variables, both will be introduced in these notes. 2 Discrete random variables 2.1 Introduction Examples of discrete random variables: Plan to roll a die 4 times, let X be the number of rolls we will see where the die showed an even number. The possible values for X are 0 , 1 , 2 , 3 , 4. Let X be number of traffic tickets you will get during the next year. The possible values for X are 0 , 1 , 2 , · · · Plan to randomly select an individual from a population. Let X = 0 be the event that the individual is Male and let X = 1 be the event that this individual is Female. The possible values of X are 0,1. Definition: probability mass function (pmf) The pmf of a discrete random variable X is the function p ( x ), p ( x ) = P ( X = x ) defined for all -∞ < x < . All discrete random variables have a pmf.
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