Random Variables Discrete variables Numerical values associated with elements in a sample space. Only distinct (discrete) points on the number line. – Probability is distributed to the discrete set of values that a random variable can take on.

The Deal - Continued Bag o’ chips (poker chips). Some are red. Some are white. Some are blue. Draw a chip from the bag. The sample space is the bag of chips. We now know that 12 are white, 6 are blue and 2 are red.

The Deal - Continued Draw a blue chip get 2 Extra Credit points. Draw a red chip get 1 Extra Credit point. Draw a white chip lose 2 Extra Credit points. The color of the chip is related to a numerical value, the number of bonus points won or lost. Therefore, the number of bonus points won or lost is a random variable with a discrete set of possible values.

Discrete Random Variable X = number of bonus points x –2 +1 +2 P(x) 0.60 0.10 0.30 The probabilities come from the fact that there are 12/20 white chips (each worth –1 points), 6/20 blue chips (each worth +3 points) and 2/20 red chips (each worth +1 points).

Discrete R.V. X = number of bonus points The distribution of probability for a discrete random variable can be displayed as a bar chart with the height of the bar equal to the probability associated with that particular value.

Discrete R.V. Property 1 Property 2 Probabilities are between 0 and 1 and the total probability associated with all the possible values of a random variable is 1.

Mean of a Discrete R.V. The “center” of the distribution of values found as a weighted average of the values. The mean is sometimes called the expected value. The mean of a random variable is the average value for that random variable.

Discrete Random Variable X = number of bonus points x –2 +1 +2 P(x) 0.60 0.10 0.30 xP(x) –1.2 +0.10 +0.60 μ =–0.5 The fact that the mean is 0.4 means that in the long run people playing this game of drawing a chip are going to come out ahead, on average. If the mean were zero, people would just be trading points. What some students lose is distributed to the students who win. In once sense this is a “fair” game because no one has an advantage in the long run.

Variance of a Discrete R.V. The “spread” of the distribution of values found as a weighted average of the squared deviations from the mean. The variance of a random variable measures spread in a way analogous to the way the sample variance measures spread for a sample.

Discrete Random Variable X = number of bonus points x –2 +1 +2 (x-μ)2 (–1.5)2 (+1.5)2 (+2.5)2 P(x) 0.60 0.10 0.30 (x-μ)2 P(x) +1.35 +0.225 +1.875 σ2=3.45 The variance of a random variable is more meaningful when comparing two or more random variables. For example, the random variable that counts the number of heads in two flips of a fair coin has the following distribution: x: 0 1 2 P(x): 0.25 0.50 0.25 The mean of this distribution is 1 and the variance of this distribution is 0.5. Therefore, this distribution is less spread out than the one above.

Rake it in!TM x P(x) $0 1,115,988/1,440,000 $1 192,000/1,440,000 $2 72,000/1,440,000 $3 28,800/1,440,000 $7 19,200/1,440,000 $14 7,200/1,440,000 $31 4,800/1,440,000 $1,500 12/1,440,000 On August 27, 2007 the Iowa Lottery introduced a new scratch game called Rake it in! The Iowa Lottery printed 1,440,000 scratch tickets for the game. The discrete random variable x is the amount a holder of one of these scratch tickets wins. Note that 192,000 tickets will win the $1 it cost to buy the ticket. There are only 12 tickets that have the top prize of $1,500.

Rake it in!TM If all 1,440,000 tickets are sold and if all prizes are claimed, the Iowa Lottery will payout $824,400 Mean payout μ = $824,400/1,440,000 = $0.5725 This means the Iowa Lottery pays out, on average, under 60 cents for every $1 ticket sold. The mean payout is $0.5725 or less than 60 cents. Because the Iowa Lottery gets $1 for every ticket sold, it makes, on average, over 40 cents on each scratch ticket. So the Iowa Lottery is the one who is raking it in.