7.2 Day 1: Mean & Variance of Random Variables Law of Large Numbers

The Mean of a Random Variable The mean x of a set of observations is their ordinary average, but how do you find the mean of a discrete random variable whose outcomes are not equally likely? The Mean of a Random Variable is known as its expected value.

Ex 1: The Tri-State Pick 3 In the Tri-State Pick 3 game that New Hampshire shares with Maine and Vermont, you choose a 3-digit number and the state chooses a 3-digit winning number at random and pays you $500 if your number is chosen.

Since there are 1000 possible 3 digit numbers, your probability of winning is 1/1000.

The probability distribution of X (the amount your ticket pays you) Payoff X:$0$500 Probability: The ordinary average of the two possible outcomes is $250, but that makes no sense as the average because $0 is far more likely than $500. In the long run, you would only receive $500 once in every 1,000 tickets and $0 in the remaining 999 of the tickets

So what is the mean? The long-run average payoff or mean for this random variable X is fifty cents. This is also known as the Expected Value. We will say that μ x = $0.50.

Mean of a Discrete Random Variable Suppose that X is a discrete random variable whose distribution is Value of X: x 1 x 2 x 3 … x k Probability: p 1 p 2 p 3 … p k To find the mean of X, multiply each possible vlaue by its probability, then add all the products  μ x = x 1 p 1 + x 2 p 2 + … + x k p k  = Σx i p i We will use μ x to signify that this is the mean of a random variable and not of a data set.

Ex 2: Benford’s Law Calculating the expected first digit What is the expected value of the first digit if each digit is equally likely? μ x = 1(1/9) + 2(1/9) + 3(1/9) + 4(1/9) + 5(1/9) + 6(1/9) + 7(1/9) + 8(1/9) + 9(1/9) = 5 First Digit X Probability1/9 The expected value is μ x = 5.

What is the expected value if the data obeys Benford’s Law? μ x = 1(.301) + 2(.176) + 3(.125) + 4(.097) + 5(.079) + 6(.067) + 7(.058) + 8(.051) + 9(.046) = First Digit X Probability The expected value is μ x =

Probability Histogram for equally likely outcomes 1 to 9 In this uniform distribution, the mean 5 is located at the center.

Probability Histogram for Benford’s Law The mean is in this right skewed distribution.

Recall… Computing a measure of spread is an important part of describing a distribution (SOCS) The variance and the standard deviation are the measures of spread that accompany the choice of the mean to measure center.

Variance of a Discrete Random Variable Suppose that X is a discrete random variable whose distribution is Value of X: x 1 x 2 x 3 … x k Probability: p 1 p 2 p 3 … p k And that the mean μ is the mean of X. The variance of X is σ x 2 = (x 1 – μ x ) 2 p 1 + (x 2 – μ x ) 2 p 2 + … + (x k – μ x ) 2 p k The standard deviation σ x of X is the square root of the variance. We will use σ x 2 to signify the variance and σ x for the standard deviation.

Ex 3: Linda Sells Cars Linda is a sales associate at a large auto dealership. She motivates herself by using probability estimates of her sales. For a sunny Saturday in April, she estimates her car sales as follows: Cars Sold: 0123 Probability:

Find the mean and variance. μ x = 1.1 σ x 2 = xixi pipi xipixipi (x i – μ x ) 2 p i (0 – 1.1) 2 (0.3) = (1 – 1.1) 2 (0.4) = (2 – 1.1) 2 (0.2) = (3 – 1.1) 2 (0.1) = The standard deviation is σ x = 0.943

The Law of Large Numbers Draw independent observations at random from any population with finite mean μ. Decide how accurately you would like to estimate μ. As the number of observations drawn increases, the mean x of the observed values eventually approaches the mean μ of the population as closely as you specified and then stays that close.

Ex 4: Heights of Young Women (Law of Large Numbers) The average height of young women is 64.5 in.

The Law of Small Numbers The law of small numbers does not exist, although psychologists have found that most people believe in the law of small numbers. Most people believe that in the short run, general rules of probability with be consistent. This is a misconception because the general rules of probability only exist over the long run. In the short run, events can only be characterized as random.

How large is a large number? The law of large numbers does not state how many trials are necessary to obtain a mean outcome that is close to μ. The number of trials depends on the variability of the random outcomes. The more variable the outcomes, the more trials that are needed to ensure that the mean outcome x is close the distribution mean μ.