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Discrete Random Variables
Discrete list of distinct values.
Typically used when a variable is integer-valued
without too many possible choices. choices
pdf (Probability distribution
function)
For example: P(X = 2) = 0.278
If X is the sum of two fair dice when rolled.
Continuous Random Variable
Assumes a range of values covering an interval.
_____________.
May be limited by instrument’s accuracy / decimal points,
but still continuous.
is this
area
Find probabilities using a
probability density function,
which is a curve.
Calculate probabilities by
finding the area under the
curve.
• We can’t find probabilities for exact outcomes.
• For example: P(X = 2) = 0.
• Instead we can find probabilities for a range of
values.
3
Expected Value = Sum of “value × probability”
over all possible values
Faculty Example: Calculate the Expected
Value E(X)
k
P(X = k)
0
0.1
1
0.3
2
0.4
3
0.2
Total
1.0
X = # courses/semester taught by
PSU faculty
 E(X) = µ =
0×(.1) + 1×(.3) + 2×(.4) +
3×(.2)
= 1.7
Interpretation:
• The average is 1.7 classes for this
population
Conditions for a
binomial experiment
1
There are n “trials”, where n is fixed and
known in advance
2
We can define two possible outcomes for
each trial: “Success” (S) and “Failure” (F)
3
The outcomes are independent; no single
outcome influences any other outcome
4
The probability of “Success” is the same for
each trial. We use “p” to write P(Success).
Mean and standard deviation for
binomial random variables
Mean:
Standard deviation:
How to relate all
this to Z-scores
• We can standardize
values from any normal
distribution to relation
them to the standard
normal distribution.
Value  Mean
z
Standard Deviation