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# Means and variances of random variables

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### Means and variances of random variables

1. 1. Means and Variances of Random Variables
2. 2. Means and Variances of Random Variables The Expected Value of a Random Variable E(X) = π π₯ = mean of a random variable = π₯ π π π Ex. Rolling 2 dice X- outcomes P- probability outcome occurs X 2 3 P 1 2 3 4 5 6 5 4 3 36 36 36 36 36 36 36 36 36 π₯π ππ 4 5 6 7 8 9 10 11 12 2 1 36 36 2 6 12 20 30 42 40 36 30 22 12 36 36 36 36 36 36 36 36 36 36 36 E(X) = π π₯ = 252 36 =7 Therefore the expected outcome or the mean outcome of rolling 2 dice is 7
3. 3. Variances of Random Variables Variance of Random Variables: Var(X) = π 2 = (π₯ π β π π₯ )2 β π π Ex. The probability of selling X number of cars is given in the table below. Find the standard deviation for selling X cars. Number of Cars (X) Probability of X 0 0.3 1 0.4 2 0.2 3 0.1 Calculate π π₯ π₯π ππ 0 0.4 0.4 0.3 (π₯ π β π) -1.1 -0.1 0.9 1.9 (π₯ π β π)2 1.21 0.01 0.81 3.61 (π₯ π β π)2 π π 0.363 0.004 0.162 0.361 π 2 = (π₯ π β π π₯ )2 β π π = 0.89 Standard Deviation = π = 0.89 = 0.943 = π. π
4. 4. Rules for Means and Variances of Random Variables When you add or subtract 2 random variables, what happens to their mean and variance? Rules: π π₯+π¦ = π π₯ + π π¦ π π₯βπ¦ = π π₯ β π π¦ π2 = π2 + π2 π₯+π¦ π₯ π¦ π2 = π2 + π2 π₯βπ¦ π₯ π¦ When you multiply each outcome of a random variable by a number, what happens to the mean and variance? Rules: π ππ₯ = ππ π₯ π 2 = π2 π 2 ππ₯ π₯ Note: Variance is multiplied by a squared factor since the spread is being multiplied in 2 directions. When you add a number to each outcome of a random variable, what happens to the mean and the variance? Rules: π π+π = π + π π₯ π2 = π2 π₯ π+π Note: Variance doesnβt change because the spread of the data remains the same