6.1 Random Variables
Def’n: A random variable is a numerical measurement of the outcome of a random
A discrete random variableis a random variable that assumes separate values.
e.g. # of people who think stats is dry
The probability distribution of a discrete random variable lists all possible values
that the random variable can assume and their corresponding probabilities.
Notation: X = random variable; x = particular value;
P(X = x) denotes probability that X equals the value x.
Ex6.1) Toss a coin 3 times. Let X be the number of heads.
8 possible values: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT
x P(X = x)
Two noticeable characteristics for discrete probability distribution:
1. 0 ≤ P(X = x) ≤ 1 for each value of x
2. ∑ P(X = x) = 1
Ex6.2) “no heads”: P(X = 0) = 0.125
“at least one head”: P(X ≥ 1) = P(X = 1) + P(X = 2) + P(X = 3)
= 0.375 + 0.375 + 0.125 = 0.875
“less than 2 heads”: P(X < 2) = P(X = 0) + P(X = 1) = 0.125 + 0.375 = 0.500
The population mean µ of a discrete random variable is a measure of the center of its
distribution. It can be seen as a long-run average under replication. More precisely,
µ = x P(X = x )
∑ i i
Sometimes referred to as µ = E(X) = the expected value of X (see STAT 265).
Keep in mind that µ is not necessarily a “typical” value of X (it’s not the mode).
Ex6.3) Using Table 6X0,
µ = ∑ x i(X = x i = (0)(0.125) + (1)(0.375) + (2)(0.375) + (3)(0.125) = 1.5
Æ On average, the number of heads from 3 coin tosses is 1.5.
Ex6.4) Toss an unfair coin 3 times (hypothetical). Let X be as in previous example.
x P(X = x) µ = ∑ xiP(X = x i
1 0.05 = (0)(0.10) + (1)(0.05) +
(2)(0.20) + (3)(0.65)
2 0.20 = 2.4
3 0.65 As 2 example shows, interpretation of µ as a measure of center of a distribution is more
useful when the distribution is roughly symmetric, less useful when the distribution is
The population standard deviation σ of a discrete random variable is a measure of
variability of its distribution. As before, the standard deviation is defined as the square
root of the population variance σ , given by
σ = ∑ (xi− µ) P(X = x ) i ∑ xiP(X = x ) i µ 2
Def’n: A continuous random variable assumes any value contained in one or more
e.g. average alcohol intake by a student, average alcohol outtake by a student
The probability distributionof a continuous r.v. is specified by a curve.
Two noticeable characteristics for continuous probability distribution (sans calculus):
1. The probability that X assumes a value in any interval lies in the range 0 to 1.
2. The interval containing all possible values has probability equal to 1, so the total area
under the curve equals 1.
(corresponding diagrams drawn in class)
Using probability symbols, point 1 is denoted by
P(a ≤ X ≤ b) = Area under the curve from a to b
(diagram drawn in class)
The probability that a continuous random variable X assumes a single value is always
zero. This is because the area of a line, which represents a single point, is zero.
(diagram drawn in class)
In general, if a and b are two of the values that X can assume, then
P(a) = 0 and P(b) = 0
Hence, P(a ≤ X ≤ b) = P(a < X < b). For a continuous probability distribution, the
probability is always calculated for an interval. Either like abov