Statistical Inference
Mr. Rizos
Linear Combinations of Random Variables
A random variable is a function that assigns a number to each outcome in the sample space ε.
A random variable whose set of values is countable, i.e., can be listed as a sequence of numbers,
is called a discrete random variable.
For any discrete probability function p(x), the following conditions must hold:
• Each value of p(x) is in the interval [0, 1]. That is, 0 ≤ p(X) ≤ 1 for all x.
P
• The sum of all the values of p(x) must be 1. That is,
p(x) = 1.
x
Consider a random variable X. We can scale this random variable by a parameter a and
shift its location by another parameter b to obtain a new random variable Y = aX + b.
This process allows us to determine probabilities associated with Y by using the original
probability distribution X.
Expected Value of a Discrete Random Variable
The expected value of a discrete random variable, X, is denoted by E(X). It is also referred
to as the mean of X, which is denoted by the symbol µ (Greek ‘m’).
X
µ = E(X) =
xn p (xn )
n
Expected values have the following linearity property: E(aX + b) = a E(X) + b.
Variance of a Discrete Random Variable
Variance is an informal measure of a distribution’s spread with respect to the mean. A
small variance implies that the possible values of a distribution are close to the mean. The
variance of a random variable, X, is given by Var(X), where:
X
Var(X) = E (X − µ)2 =
(xn − µ)2 p(xn )
n
The following properties are useful for calculation purposes:
Var(X) = E(X 2 ) − µ2
Var(ax + b) = a2 Var(X)
Standard Deviation of a Discrete Random Variable
The Standard Deviation, σ, of a random variable, X, is the positive square root of the
Variance. That is,
p
σ = SD(X) = Var(X)
A useful property of the standard deviation is that it is expressed in the same units as the
data values, and hence can be used to determine margins of error and confidence intervals
in experiments.
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Jim Rizos
Examples
1. The probability distribution of X, the number of purchases that person makes at a particular online store per month is given in the following table.
Number of online orders, x
Pr(X = x)
0
0.2
1
0.4
2
a
3
0.1
4
0.1
The online store has a $5 per month membership fee as well as a flat rate $12 express post
delivery fee per online purchase.
(a) Determine the value of a.
(b) Express the person’s monthly spending, S, at this online store as a linear function of
X.
(c) Use your answer from part b. to complete the table below.
Monthly online spending, s
Pr(S = s)
(d) Determine the probability that the person spends more than $25 per month at this
online store.
(e) Determine E(S) and Var(S).
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Jim Rizos
2. If E(X) = 4 and Var(X) = 3, find:
(a) E(2X + 1)
(b) Var(2X + 1)
(c) SD(2X + 1)
Continuous Random Varibales
If X is a continuous random variable with probability density function f (x), then:
• f (x) ≥ 0 for any real number x.
Z ∞
•
f (x) dx = 1
−∞
Z
x2
• Pr (x1 ≤ X ≤ x2 ) =
f (x) dx
x1
• Pr(X = x) = 0 for any real number x.
Expected Value of a Continuous Random Variable
Z
∞
xf (x) dx
Mean = µ = E(X) =
−∞
Variance and Standard Deviation of a Continuous Random Variable
Var(X) = E (x − µ)
2
Z
∞
Z
2
∞
(x − µ) f (x) dx =
=
−∞
−∞
σ = SD(X) =
3
p
Var(X)
x2 f (x) dx − µ2
Jim Rizos
Examples
2. The random variable X has the following probability density function:
(
a x2 − 2x − 3 if 0 ≤ x ≤ 3
f (x) =
0
elsewhere
Find the following statistics:
(a) E(X)
(b) Var(X)
(c) SD(X)
(d) Pr(X ≤ 2)
Consider another random variable, Y = 3X + 1.
(e) Determine Pr(Y ≥ 7).
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Jim Rizos
A Linear Combination of Random Variables
For independent random variables X and Y and constants a and b:
• E(aX + bY ) = aE(X) + bE(Y )
• Var(aX + bY ) = a2 Var(X) + b2 Var(Y )
Examples
4. To get to university, a student takes two different buses. The time travelled on the first
bus, X minutes, is a continuous random variable with a mean of 25 minutes and standard
deviation of 5 minutes. The time travelled on the second bus, Y minutes, is a continuous
random variable with a mean of 15 minutes and standard deviation of 3 minutes.
Find the mean and standard deviation of the total time taken for the student to get to
university, if the times taken for each part of the journey are independent.
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