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204 6.1 random variables

# 204 6.1 random variables - has a countable number of...

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6.1 Discrete Random Variables Assume we perform the experiment of rolling two distinct Dice and observe the number of dots on the top face. The Sample space for this experiment is: (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) The cardinality of this sample space is 36 A random variable is a function on the sample space i.e. it is the assignment of values to the possible outcomes of an experiment. For example we can perform the experiment of tossing two dice and observing the number of dots on the top face of each die obtaining the above sample space. One possible random variable which can be assigned to this sample space is the sum of the number of dots showing on each of the dice. This is an example of a discrete random variable, that is one which

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Unformatted text preview: has a countable number of different values to which the random variable can be assigned. In the two dice case this number is 11 (2,3,4,5,6,7,8,9,10,11,12). Some random variables have an uncountable number of possible values (examples are the weight of a person or length of time spent waiting) in which the possible values range continuously over the real numbers. These random variables are called continuous random variables. For the random variable of the sum of the number of dots on the top faces we obtain the following probability distribution: X 2 3 4 5 6 7 8 9 10 11 12 P(X) Expected value and variance for a Discrete Random Variable (RV) Expected Value E(X) = ∑ x p(x) Variance VAR(X) = ∑-2 ) ( μ x p(x) = 2 2 ) ( ) (-∑ x p x Example: X 1 2 3 4 P(X) .1 .3 .2 .1 .3 E(X) VAR(X)...
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