1. Chapter 16: Random Variables
Lajja Majmundar & Vishmi Abeygunarathna

2. Vocabulary
Random Variable: A random variable assumes any of the several different values as a result of some random event. Random variables are denoted by a capital letter such as X.
Discrete Random Variable: A random variable that can take one of a finite number of distinct outcomes is called a discrete random variable.
Continuous Random Variable: A random variable that can take any numerical value within a range of values is called a continuous random variable. The range may be infinite or bounded at either or both ends.

3. Probability Model: The probability model is a function that associates a probability P with each value of a discrete random variable X, denoted P(X=x), or with any interval of values of a continuous random variable.
Expected Value: The expected value of a random variable is its theoretical long-run average value, the center of its model. Denoted μ or E(x), it is found (if random variable is discrete) by summing the products of variable values and probabilities.
Variance: The variance of a random variable is the expected value of the squared deviation from the mean.
Standard Deviation: The standard deviation of a random variable describes the spread in the model, and is the square root variance.

4. Formulas Expected Value: E(X) = x1p1 + x2p2 + x3p3 + . . . + xnpn
Variance:
σ2 = Var(X) = Σ(x-μ)2P(x)
Standard Deviation:
σ = SD(X) = √Var(X)
Adding/Subtracting Constants:
E(X +/- c) = E(X) +/- c
Var(X +/- c) = Var(X)

5. Multiplying Variables:
E(aX) = aE(X)
Var(aX) = a2Var(X)
Adding/Subtracting Variables:
E(X +/- Y) = E(X) +/- E(Y)
if X and Y are independent
Var(X +/- Y) = Var(X) + Var(Y)

6. Main Concepts
A random variable is a value based on the outcome of a random event
outcomes can either be discrete or continuous based on how they are listed
The collection of all possible values and the probabilities that they occur is called the probability model for the random variables
The expected value can be calculated by multiplying each possible value by its probability and then finding the sum
The variance is the expected value of the squared deviations
Adding or subtracting a constant from data shifts the mean but does not change the variance or standard deviation
The same is true for random variables

7. Multiplying each variable by a constant multiplies the mean by that constant and the variance by the square of that constant
The expected value of the sum is the sum of the expected values
The variance of the sum of two independent random variables is the sum of their individual variances
The mean of the sum of two random variables is the sum of the means
The mean of the difference of two random variables is the difference of the means
If the random variables are independent, the variance of their sum of difference is always the sum of the variances
When two independent continuous variables have Normal models, so does their sum or difference

8. HW Problem #23
Random Variables. Given independent random variables with means and standard deviation as shown, find the mean and standard deviation of each of these variables.
a.) 3X
b.) Y+ 6
c.) X + Y
d.) X - Y
e.) X1 + X2
Mean SD X 10 2 Y 20 5

9. a.) μ = E(3X) = 3E(X) = 3(10) = 30
σ = SD(3X) = 3SD(x) = 3(2) = 6
b.) μ = E(Y + 6) = E(Y) + 6 = = 26
σ = SD(Y + 6) = SD(Y) = 5
c.) μ = E(X + Y) = E(X) + E(Y) = = 30
σ = SD(X + Y) = sqrt(Var(X) + Var(Y)) = sqrt( ) = 5.39
d.) μ = E(X - Y) = E(x) - E(Y) = = -10
σ = SD(X - Y) = sqrt(Var(X) + Var(Y)) = sqrt( ) = 5.39
e.) μ= E(X1 + X2) = E(X) + E(X) = = 20
σ = SD(X1 + X2) = sqrt(Var(X) +Var(X)) = sqrt( ) = 2.83

10. HW Problem #25
Random Variables. Given independent random variables with means and standard deviation as shown, find the mean and standard deviation of each of these variables.
a.) 0.8Y
b.) 2x - 100
c.) X + 2Y
d.) 3X - Y
e.) Y1 + Y2
Mean SD X
120 12 Y
300 16

11. a.) μ = E(.8Y) = .8E(Y) = .8(300) = 240
σ = SD(.8Y) = .8SD(Y) = .8(16) = 12.8
b.) μ = E(2X - 100) = 2E(X) = 140
σ = SD(2X - 100) = 2SD(X) = 2(12) = 24
c.) μ = E(X + 2Y) = E(X) + 2E(Y) = (300) = 720
σ = SD(X + 2Y) = sqrt(Var(X)+ 22Var(Y)) = sqrt( (162) = 34.18
d.) μ = E(3X - Y) = 3E(X) - E(Y) = 3(120) = 60
σ = SD(3X - y) = sqrt(32Var(X)+ Var(Y)) = sqrt(32(122) + 162) = 39.40
e.) μ = E(Y1 + Y2) = sqrt(Var(Y) + Var(Y)) = sqrt( ) = 22.63