University:
California State University, Northridge
Course:
MATH 140 | Introductory Statistics
Academic year:

2013
Views:

174

Pages:

15
Author:

Emily Clayton

Variances of Probability distributions σ 2 n =∑ SD = σ (x − x) 2 n 2 n = ∑ (x − x) 2 n For a set of data - note the n instead of n-1 € Variances of Probability distributions For a probability distribution P(X) σ 2 x =∑ SD = σ (x − x) 2 n 2 n = ∑ (x − x) n 2 Try calculating the variance and SD Recall, the average is 60.14 cents or 0.6014 dollars Try calculating the variance and SD The way to think about this is: for each deviation from the mean, what is the probability? For example The contribution for the $1 winning is (1 - 0.6014)2 * 1/10 Try calculating the variance and SD So all together, the variance is (1 - 0.6014)2 * 1/10 + (2 - 0.6014)2 * 1/14 + etc etc =16.32 The standard deviation is its square root = $4.04 Variances Significance The average payout is $0.60 With a SD of $4.04 This means on average you win little but there is a chance of winning a lot (that is why the SD is large) Let’s triple the lottery Calculate mean and standard deviation Compare to original lottery Let’s triple the lottery Original lottery winnings $0.6014 New winnings $1.804 Is this smart on Wisconsin’s part? Original SD = $4.04 New SD = $12.12 How are they related? They are both multiplied by 3! SD = σX Let’s triple the lottery µx turns into 3 µx σX turns into 3 σX This is true in general. Multiplying data will rescale average and SD of the distribution What if I had decided to add 50 cents to each winning? Let’s add 50 cents to the payout µx turns into 0.50+ µx σX stays σX Just like rescaling and recentering So, adding C and multiplying by D gives µx turns into µC+DX = C+ D*µx σX turns into σC+DX = D*σX In general Now, this was for TRIPLING the lottery What if we kept the same lottery and bought 3 tickets? What do you think? If every time I play my average payout is $0.6014 What do I get after buying 3 tickets? Duh - 3 * 0.6014 = 1.804! Just like before! It does not matter if I triple the lottery or if I buy three tickets, the result is the same. My take-home on average is tripled. What do you think? We can conclude that when we select three items from the same distribution we find µ 3,X = 3 µx = µx + µx + µx In general, for different distributions we get µ X,Y = µx + µy Two tickets from two lotteries Let’s buy a ticket from the lottery of California and of Texas California µx = $0.50 Texas µY = $0.75 What are the expected total winnings? µ CA,TX = µCA + µTX = $0.50 + $0.75 = $1.25