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This document introduces random variables, focusing on discrete cases before extending to continuous ones. A random variable is a numerical outcome of an experiment; its probability distribution shows the likelihood of each value. The probability mass function, p(x), maps each value to its probability, summing to one. The text distinguishes between random variables (numeric values) and events (occurrences). Expected value E(X) is calculated by summing each value multiplied by its probability, representing the average outcome over many trials. Higher moments, like E(X²), are also introduced, crucial for calculating variance. Variance, measuring the spread of the distribution, is defined as E[(X - E(X))²] = E(X²) - [E(X)]². Standard deviation is the square root of the variance. The moment generating function, MX(t) = E(e^(Xt)), provides a method for easily calculating moments through differentiation. The document then extends these concepts to continuous random variables, replacing sums with integrals and introducing the probability density function f(x), where the integral of f(x) over the range equals 1. Examples using coin flips and dice rolls illustrate the calculations of expected value, variance, and standard deviation, along with the application of the moment generating function.
Random Variables, Expected Value, Variance and Moments
Kevin Burke University of Limerick, Maths & Stats Dept 1 / 31
Random Variables Expected Value Variance and Standard Deviation Moment Generating Function Continuous RVs
Random Variables
In probability theory, a random variable is a numerical quantity whose
value is determined by an experiment.
For example, consider the experiment of flipping two coins.
Now define a random variable X = “the number of heads” whose value
will clearly be 0, 1 or 2 heads:
Outcome HH HT TH TT
Value assigned to X 2 1 1 0
Kevin Burke University of Limerick, Maths & Stats Dept 2 / 31
Random Variables Expected Value Variance and Standard Deviation Moment Generating Function Continuous RVs
Distribution of a Random Variable
The probability distribution of X is:
x 0 1 2
Pr(X = x)
1
4
1
2
1
4
This describes how likely each of the values are, i.e., how the
probability gets distributed to each possible value of X.
Note that upper case X denotes the random variable whereas lower
case x represents a specific value.
Pr(X = x) means “the probability that the random variable X attains
the specific value x” where x ∈ {0, 1, 2}, e.g., Pr(X = 0) = 1
4
.
Kevin Burke University of Limerick, Maths & Stats Dept 3 / 31
Random Variables Expected Value Variance and Standard Deviation Moment Generating Function Continuous RVs
Probability Mass Function
Pr(X = x) is called the probability mass function - it maps each
value of X to a probability value.
This is often shortened to p(x) - pronounced “p - of - x”.
The probability values of this function must sum to one:
Xp(xi) = 1 .
In the previous example, p(0) = 1
4
, p(1) = 1
2
and p(2) = 1
4
.
⇒ p(0) + p(1) + p(2) = 1.
Kevin Burke University of Limerick, Maths & Stats Dept 4 / 31
Random Variables Expected Value Variance and Standard Deviation Moment Generating Function Continuous RVs
Random Variable Vs Event
Previously we encountered events - not the same as random variables.
For the sake of clarity consider:
1
4
(1 + 2 e
t + e
2t
).
Differentiating this twice gives
d
dt MX (t) = 1
4
(2 e
t + 2 e
2t
),
d
2
dt2 MX (t) = 1
4
(2 e
t + 4 e
2t
),
and the first two moments are
E(X) = d
dt MX (0) = 1
4
(2 e
0 + 2 e
0
) = 4
4 = 1,
E(X
2
) = d
2
dt2 MX (0) = 1
4
(2 e
0 + 4 e
0
) = 6
4 = 1.5,
as previously calculated.
Kevin Burke University of Limerick, Maths & Stats Dept 23 / 31
Random Variables Expected Value Variance and Standard Deviation Moment Generating Function Continuous RVs
Question 4
We had the random variable Y = “the number of unique faces” based
on flipping a coin twice.
y 1 2
Pr(Y = y)
1
2
1
2
a) Derive the moment generating function.
b) Use the answer to part (a) to calculate E(X) and E(X
2
).
Kevin Burke University of Limerick, Maths & Stats Dept 24 / 31
Random Variables Expected Value Variance and Standard Deviation Moment Generating Function Continuous RVs
Discrete Versus Continuous
We have considered discrete random variables:
X takes values from a discrete set {x1, x2, x3, . . . , xn}.
The total probability is Pr(X ∈ {x1, x2, x3, . . . , xn}) = Pp(xi) = 1.
Here p(x) = Pr(X = x) is the probability mass function which
assigns a probability to each value.
Now consider continuous random variables:
X can take any value in an interval [a, b].
The total probability is Pr(X ∈ [a, b]) = 1.
We cannot assign a probability to each X ∈ [a, b] since there are an
infinite number of values - so what can we do?
Kevin Burke University of Limerick, Maths & Stats Dept 25 / 31
Random Variables Expected Value Variance and Standard Deviation Moment Generating Function Continuous RVs
Understanding Continuous Variables
To bridge the gap between discrete and continuous distributions, we
can split [a, b] into n − 1 discrete sub-intervals using n equally spaced
points: a = x1 < x2 < · · · < xn = b.
⇒ Pr(X ∈ [a, b] ) =
nX−1
i=1
Pr(X ∈ [xi
, xi+1] ) = 1.
Now assume that there exists a function f(x) which describes these
n − 1 probabilities as follows:
Pr(X ∈ [xi
, xi+1] ) = f(xi)∆x
Here ∆x = xi+1 − xi
is the distance between the equally spaced
points. Note: since Pr(X ∈ [xi
, xi+1] ) cannot be negative, f(x) ≥ 0.
Kevin Burke University of Limerick, Maths & Stats Dept 26 / 31
Random Variables Expected Value Variance and Standard Deviation Moment Generating Function Continuous RVs
Every Value in the Continuous Interval
Thus, the previous sum becomes
Pr(X ∈ [a, b] ) =
nX−1
i=1
f(xi)∆x = 1.
To incorporate every value in [a, b], we must increase n to infinity
producing an infinite number of sub-intervals of length ∆x ≈ 0.
Thus we get
Pr(X ∈ [a, b] ) = lim
n→∞
Xn−1
i=1
f(xi)∆x =
Z b
a
f(x) dx
| {z }
By definition of an integral
= 1.
Kevin Burke University of Limerick, Maths & Stats Dept 27 / 31
Random Variables Expected Value Variance and Standard Deviation Moment Generating Function Continuous RVs
Continuous: Probability Density Function
Probabilities are calculated (using integration) through a probability
density function which has the following properties:
1
18
e
6t
(6t − 1) + 1
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