Download Standard Normal Random Variable

Section 7.1 Properties of the Normal Distribution
Objective: State the properties of the normal curve; interpret the area under
a normal curve.
 To find probabilities for discrete random variables, we used probability
distribution functions (Section 6.2).
 To find probabilities for continuous random variables, we use probability
density functions.
o The total area under the curve of the graph of the function must equal 1.
o The area under the curve represents the probability of observing the
random variable in the given interval.
o The height of the graph must be greater than or equal to zero.
o The probability density function for a uniform random variable is a
rectangle because all possible values of the random variable in an interval
are equally likely.
o If the relative frequency histogram is symmetrical and bell-shaped, then
it is a normal curve.
7.1 - 2
Fig 7-1: A Normal Curve
The graph of a normal distribution
is called a normal (bell-shaped) curve.
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7–2
Important Properties of a Normal Curve:
 Highest point over the mean, .
 Symmetrical about a vertical line through .
 The curve approaches the horizontal axis but never touches or crosses it.
 The transition (inflection) points between cupping upward and
downward occur at  +  and  - .
 As x becomes extremely large or extremely small, the graph approaches
but never reaches the horizontal axis.
Work #1 - 4
Standard Normal Random Variable
To find the area under any normal curve, we find the Z-score of the normal
random variable, then use a table (or graphing calculator) to find the area.
Z
X 

The area under the curve between two normal random variables is the same as
the area under the curve between their respective Z-scores.
Work #5