Download Random Variables Probability mass functions

Math 10B Section
Worksheet # 13
Author: Bo Lin
Random Variables
Definition 1. Let Ω be a probability space. A random variable is a function
X : Ω → R.
In other words, the random variable X assigns a real number X(ω) to each
possible outcome ω ∈ Ω.
Remark 2.
• Keep in mind that a random variable is a function.
• The motivation of introducing random variables is we want to measure
the complicated phenomena we observe. So X is a link from the real world
(those ω) to probability theory (X(ω)).
Probability mass functions
Definition 3. The range of a random variable X is just its range as a function.
X is called discrete if its range R is a finite set or an infinite discrete set and
P (X = x) > 0
for all x ∈ R.
Theorem 4. If f is the probability mass function of a random variable X with
range R, then
X
X
f (x) =
P (X = x) = 1.
x∈R
x∈R
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Examples
1. Suppose you draw one card randomly from the standard 52-card deck. X
is the random variable of the number of black card you draw. What is the
probability mass function of X? (among the four suits, spades and clubs are
black, hearts and diamonds are red)
2. Suppose you flip a fair coin 3 times. Let X be the number of pairs of
consecutive heads. For example if the outcome is HHH, then X = 2. What is
the range of X? What is the probability mass function of X?
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