Discrete random variables
Malabika Pramanik
Math 105 Section 203
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Math 105 (Section 203)
Discrete random variables
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Random variable
Definition
A random variable X is a variable whose value depends on chance.
Example:
1
Suppose you toss a coin. You gain a dollar if the coin shows up
heads, otherwise you lose a dollar. Your earning from a single toss of
the coin is a random variable.
(
1
if heads
X =
−1 if tails.
2
Suppose you throw a six-faced die. The number Y that shows up on
the top face is a random variable which can take any of the values
1,2,3,4,5,6.
3
Pick a person at random from the UBC student population. The
height Z of that person is a random variable.
Math 105 (Section 203)
Discrete random variables
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Discrete and Continuous random variables
Definition
A random variable is said to be discrete if it can assume only a finite
set of values.
If a random variable can take any value in an interval, it will be called
continuous.
Examples:
1
The random variable X = ±1 in the coin-tossing experiment, the
random variable Y in the throw of a die are discrete random variables.
2
The weight variable Z is continuous.
Math 105 (Section 203)
Discrete random variables
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Probability distribution of a discrete random variable
Definition
The probability distribution of a discrete random variable is a table that
records the possible values of the random variable, together with the
relative frequency (or probability) with which they occur.
A discrete probability distribution of a random variable X is represented as
follows:
value of X
x1
x2
..
.
probability
p1
p2
..
.
xn
pn
where p1 + p2 + · · · + pn = 1.
Math 105 (Section 203)
Discrete random variables
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Example
In the experiment of tossing two dice, let X be the random variable that
gives the sum of the top faces. Which of the following gives the
probability distribution of X ?
A. xk = k, pk = 1/6, k = 1, 2, · · · , 6.
B. xk = k, pk = 1/11, k = 2, 3, · · · , 12.
C. xk = k, pk = min [k − 1, 13 − k] /36, k = 2, 3, · · · , 12.
D. xk = k, pk = k/48, k = 2, 3, · · · , 12.
Math 105 (Section 203)
Discrete random variables
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Expectation and Variance
Let X be a discrete random variable with probability distribution
value of X
x1
x2
..
.
probability
p1
p2
..
.
xn
pn
Definition
µ = E (X ) = expectation of X = x1 p1 + · · · + xn pn ,
σ 2 = Var(X ) = variance of X = (x1 − µ)2 p1 + · · · + (xn − µ)2 pn .
Math 105 (Section 203)
Discrete random variables
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Example (ctd)
Let X be the random variable in the preceding example. What is its
expected value?
A. 6
B. 7
C. 8
D. 0
E. 12
Math 105 (Section 203)
Discrete random variables
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