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Means and Variances of Random Variables

Means and Variances of Random Variables. Activity 1 : means of random Variables. To see how means of random variables work, consider a random variable that takes values {1,1,2,3,5,8}. Then do the following:. Calculate the mean of the population:

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Means and Variances of Random Variables

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  1. Means and Variances of Random Variables

  2. Activity 1: means of random Variables To see how means of random variables work, consider a random variable that takes values {1,1,2,3,5,8}. Then do the following: • Calculate the mean of the population: • Make a list of all the sample of size 2 from this population. You should have 15 subsets of size 2 • Find the mean of the 15 x-bar in the third column and compare the result with the population mean. • Repeat steps 1-3 for a different (but still small) populations of your choice. Now compare your result with each other. • Write a brief statement that describes what you discovered.

  3. Mean of a random Variable The mean of a discrete random variable X is a weighted average of the possible values that the random variable can take.

  4. example A Tri-State Pick 3 game in New Hampshire, Maine and Vermont let you choose three-digit number and the state chooses three-digit winning number at random and pays you $500 if your number is chosen. Because there are 1000 three digit numbers, you have a probability of 1/1000 of winning. Taking X to be the amount your ticket pays you, the probability distribution of X is:

  5. What is the average payoff from the tickets? ($0 + $500)/2 = $250 The long-run average pay off is: $500(1/1000) + $0(999/1000) = $0.50

  6. Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a random variable weights each outcome xi according to its probability, pi. The common symbol for the mean (also known as the expected value of X) is , formally defined by The mean of a random variable provides the long-run average of the variable, or the expected average outcome over many observations.

  7. Find the Find the y= 3.4441 x= 5 of the V distribution of the X distribution Benford’s Law Calculating the expected first digit

  8. y= 3.4441 x= 5

  9. The standard deviation is the square root of the variance. Variance of discrete Random Variables The variance of a discrete random variable X measures the spread, or variability, of the distribution, and is defined by:

  10. Gabby Sells Cars: Gabby is a sales associate at a large auto dealership. She motivates herself by using probability estimates of her sales. For sunny Saturday in April, she estimates her car sales as follows: Let’s find the mean and variance of X

  11. Law of Large number

  12. Rule for Means

  13. Rules for Variances

  14. Example: Mean of X: 1.1 cars Mean of Y: 0.7 T’s & SUV’s At her commission rate of 25% of gross profit on each vehicle she sells, Linda expects to earn $350 for each cars sold and $400 for each truck and SUV’s sold. So her earnings are: Z= 350 X + 400Y Combining rule 1 and 2 her mean earnings will be: Uz= 350 UX + 400 UY Uz= 350 (1.1)+ 400 (.7) = $665 a day

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