z-Score for a Value of a Random Variable
The z-Score for a value x of a random variable is the number of
standard deviations that x falls from the mean µ. It is calculated as
z=
x −µ
σ
Using z-Scores to Find Normal Probabilities or Random Variable x
Values:
• If we’re given a value x and need to find a probability, convert
x to a z-score using
x −µ
z=
,
σ
use a table of normal probabilities (or software, or a
calculator) to get a cumulative probability, and then convert it
to the probability of interest.
• If we’re given a probability and need to find the value of x,
convert the probability to the related cumulative probability,
and find the z-score using a normal table (or software, or a
calculator) and then evaluate
x = µ + zσ.
• Standard Normal Distribution
Then standard normal distribution is the normal
distribution with mean µ = 0 and standard deviation σ = 1. It
is the distribution of normal z-scores.
z-Score and the Standard Normal Distribution
When a random variable has a normal distribution and its values
are converted to z-scores by subtracting the mean and dividing by
the standard deviation, the z-scores have the standard normal
distribution (mean=0, standard deviation=1).
• Binomial Distribution
trial
Conditions for Binomial Distribution
• Each trial has two possible outcomes (“success” and
“failure”).
• Each trial has the same probability of a success, denoted by p.
• The n trials are independent.
The binomial random variable X is the nnumber of successes in
the n trials.
Probabilities for a Binomial Distribution
For n independent trials, the probability of x successes equals
P(x) =
n!
p x (1 − p)n−x ,
x!(n − x)!
x = 0, 1, 2, . . . , n
Here p is the probability for a success of each trial.