Means and Variances
of Random Variables
Means and Variances of Random Variables
The Expected Value of a Random Variable E(X) = πœ‡ π‘₯ =
mean of a random variable = π‘₯...
Variances of Random Variables
Variance of Random Variables: Var(X) = 𝜎 2 = (π‘₯ 𝑖 βˆ’ πœ‡ π‘₯ )2 βˆ— 𝑝 𝑖
Ex. The probability of sell...
Rules for Means and Variances of Random Variables
When you add or subtract 2 random variables, what happens to their mean ...
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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œ‡ π‘₯ = mean of a random variable = π‘₯ 𝑖 𝑝 𝑖 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‘₯𝑖 𝑝𝑖 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œ‡ 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) = 𝜎 2 = (π‘₯ 𝑖 βˆ’ πœ‡ π‘₯ )2 βˆ— 𝑝 𝑖 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œ‡ π‘₯ π‘₯𝑖 𝑝𝑖 0 0.4 0.4 0.3 (π‘₯ 𝑖 βˆ’ πœ‡) -1.1 -0.1 0.9 1.9 (π‘₯ 𝑖 βˆ’ πœ‡)2 1.21 0.01 0.81 3.61 (π‘₯ 𝑖 βˆ’ πœ‡)2 𝑝 𝑖 0.363 0.004 0.162 0.361 𝜎 2 = (π‘₯ 𝑖 βˆ’ πœ‡ π‘₯ )2 βˆ— 𝑝 𝑖 = 0.89 Standard Deviation = 𝜎 = 0.89 = 0.943 = 𝟏. 𝟏
  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œ‡ π‘₯+𝑦 = πœ‡ π‘₯ + πœ‡ 𝑦 πœ‡ π‘₯βˆ’π‘¦ = πœ‡ π‘₯ βˆ’ πœ‡ 𝑦 𝜎2 = 𝜎2 + 𝜎2 π‘₯+𝑦 π‘₯ 𝑦 𝜎2 = 𝜎2 + 𝜎2 π‘₯βˆ’π‘¦ π‘₯ 𝑦 When you multiply each outcome of a random variable by a number, what happens to the mean and variance? Rules: πœ‡ π‘Žπ‘₯ = π‘Žπœ‡ π‘₯ 𝜎 2 = π‘Ž2 𝜎 2 π‘Žπ‘₯ 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œ‡ 𝑏+𝑋 = 𝑏 + πœ‡ π‘₯ 𝜎2 = 𝜎2 π‘₯ 𝑏+𝑋 Note: Variance doesn’t change because the spread of the data remains the same

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