Chapter 16:
Random Variables
Lajja Majmundar & Vishmi Abeygunarathna
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.
● 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.
Formulas
● Expected Value:
● Variance:
E(X) = x1p1 + x2p2 + x3p3 + . . . + xnpn
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)
● 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)
Main Concepts
● A random variable is a value based on the outcome of a random event
o 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
o The same is true for random variables
● 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
o 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
o 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
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
Mean
SD
c.) X + Y
X
10
2
d.) X - Y
e.) X1 + X2
Y
20
5
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 = 20 + 6 = 26
σ = SD(Y + 6) = SD(Y) = 5
c.) μ = E(X + Y) = E(X) + E(Y) = 10 + 20 = 30
σ = SD(X + Y) = sqrt(Var(X) + Var(Y)) = sqrt(22 + 52) = 5.39
d.) μ = E(X - Y) = E(x) - E(Y) = 10 - 20 = -10
σ = SD(X - Y) = sqrt(Var(X) + Var(Y)) = sqrt(22 + 52) = 5.39
e.) μ= E(X1 + X2) = E(X) + E(X) = 10 + 10 = 20
σ = SD(X1 + X2) = sqrt(Var(X) +Var(X)) = sqrt(22 + 22) = 2.83
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
Mean
SD
c.) X + 2Y
X
120
12
d.) 3X - Y
e.) Y1 + Y2
Y
300
16
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) - 100 = 140
σ = SD(2X - 100) = 2SD(X) = 2(12) = 24
c.) μ = E(X + 2Y) = E(X) + 2E(Y) = 120 + 2(300) = 720
σ = SD(X + 2Y) = sqrt(Var(X)+ 22Var(Y)) = sqrt(122 + 22(162) = 34.18
d.) μ = E(3X - Y) = 3E(X) - E(Y) = 3(120) - 300 = 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(162 + 162) = 22.63