# Document 10505599 c

Kathryn Bollinger, June 21, 2011

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## 8.1 - Distributions of Random Variables

Random Variable ( X ) : a rule that assigns a number to each outcome of a chance experiment

Ex: Toss a fair coin two times. Let the random variable X denote the number of heads that occur.

(a) List the outcomes of the experiment and find the value assigned to each outcome by the random variable, X .

(b) Find the event comprising the outcomes to which a value of 1 is assigned by X .

Categories of Random Variables

1.

Finite Discrete: assumes a finite number of values, which can all be explicitly written

Ex: You examine each egg from a carton of a dozen eggs and notice whether each egg is broken or not. Let X = the number of broken eggs in the carton.

2.

Infinite Discrete: has infinitely many values that may be arranged in a sequence

Ex: You toss a fair coin until the coin shows tails. Let X = the number of tosses.

3.

Continuous: values comprise an interval of real numbers

Ex: Let X = the number of hours you talk on the phone in one day.

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Kathryn Bollinger, June 21, 2011

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Ex: Classify each of the following random variables and give all possible values the random variable may assume.

(a) X = the number of times you draw a card from a standard 52-card deck, with replacement, until you draw a queen

(b) X = the number of miles you travel on a single tank of gas

(c) X = the total number of hours of TV you watch on a Saturday and Sunday, recorded to the nearest hour

Probability Distributions of Random Variables

Same as previously defined, but now using the probabilities of the random variables instead of those associated with each outcome.

Ex: Let X = the number of heads that occur when tossing a fair coin two times. Find the probability distribution for X .

Histogram: a graphical representation of the probability distribution of a random variable

1. Random variable values are plotted on the x -axis.

2. Rectangles are constructed for each random variable value so that the width of each = 1 and the height of each = the probability of the random variable value occurring

=

Ex: Draw a histogram for the probability distribution you found above.