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Chapter 5 notes

# Chapter 5 notes - Chapter 5 Probability The Mathematics of...

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Chapter 5 – Probability: The Mathematics of Randomness Section 5.1: (Discrete) Random Variables Definition 1 : A random variable (rv) is a quantity that (prior to observation) can be thought of as dependent on chance phenomena. We usually use capital letters at the end of the alphabet to represent rv’s. (X,Y,Z,W, etc.) Definition 2 : A discrete random variable is one that has isolated or separated possible values (rather than a continuum of available outcomes). o Example: Let X = number of head in 10 flips of a coin. Here, the possible values for X are 0,1,2,…,10. Thus, X is discrete Definition 3 : A continuous random variable is one that can be idealized as having an entire (continuous) interval of numbers as its set of possible values. o Example: Let Y = weight of babies born at a given hospital Definition 4 : To specify a probability distribution for a rv is to give its set of possible values and consistently assign numbers between 0 and 1, called probabilities, as measures of the likelihood that the various numerical values will occur. The tool most often used to describe a discrete probability distribution is the probability mass function (pmf). Definition 5 : A probability mass function (pmf) for a discrete random variable X, having possible values x x 1 2 , , , is a nonnegative function f ( x ), with f x j ( ) giving the probability that X takes the value x j . So f ( x )=P[X = x ] where X is a random variable and x is a specific numeric value. (read as “ f ( x ) equals the probability that X equals x ) Example 5.1 : Let X=the number of goals scored by a hockey team in each of their first 9 games Suppose a team has X=1, 1, 0, 5, 0, 2, 8, 4, 1 goals. Find the function f ( x ). 1

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Graph the function f ( x ) and find P[X>2], and P[1 X 5]. Two properties necessary for a pmf: 1) f ( x ) is in the interval [0,1] for all x . 2) The values of f ( x ) sum to 1 when the sum is taken across all possible values for x . Another way of specifying a discrete probability distribution is by using the cumulative density function (cdf). Definition 6 : The cumulative probability function or cumulative density function (cdf) for a random variable X is a function F ( x ) that for each number x gives the probability that X takes that value of a smaller one. In symbols, F ( x )=P[X x ] Example 5.1 continued : Find the function F ( x ) and graph it. Also, find F (2.9). 2
Definition 7 : The mean or expected value of a discrete random variable X is E(X) = x f x x ( ) EX or E(X) is read as the “expected value of X” and is sometimes denoted by μ . Note that the summation is taken over all possible values of x . Example 5.1 continued : Find EX. Definition 8 : The variance of a discrete random variable X is Var X = ( ) ( ) ( ) ( ) x E X f x x f x E X x x - = - 2 2 2 The standard deviation of X is V a r X . Often the notation σ 2 is used in place of Var X, and σ 2 is used in place of V a r X .

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Chapter 5 notes - Chapter 5 Probability The Mathematics of...

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