Normal Probability Distribution

Section 7.1: Properties of the Normal Distribution Objectives: Students will be able to:
Understand the uniform probability distribution
Graph a normal curve
State the properties of the normal curve
Understand the role of area in the normal density function
Understand the relationship between a normal random variable and a
standard normal random variable Vocabulary:
Continuous random variable - has infinitely many values
Uniform probability distribution - probability distribution where the
probability of occurrence is equally likely for any equal length
intervals of the random variable X..
Normal curve - bell shaped curve
Normal distributed random variable - has a PDF or relative frequency
histogram shaped like a normal curve
Standard normal - normal PDF with mean of 0 and standard deviation of 1
(a z statistic!!)
Key Concepts: Probability in a Continuous Probability Distributions:
Let P(x) denote the probability that the random variable X
equals x, then
1) The sum of all probabilities of all outcomes must equal 1
S P(x) = 1
> the total area under the graph of the PDF must equal 1
2) The probability of x occurring in any interval, P(x), must
between 0 and 1 0 ? P(x) ? 1
> the height of the graph of the PDF must be greater than
or equal to 0 for all possible values of the
random variable
3) The area underneath probability density function over some
interval represents the probability of observing a value of the random
variable in that interval. [pic] Probability of a Continuous Random Variable (from a Calculus
Prospective):
x=3 x=3
( f(x) dx = 0.33 x ( = 1
x=0 x=0 Properties of the Normal Density Curve
1. It is symmetric about its mean, ?
2. Because mean = median = mode, the highest point occurs at x = ?
3. It has inflection points at ? - ? and ? + ?
4. Area under the curve = 1
5. Area under the curve to the right of ? equals the area under
the curve to the left of ?, which equals ½
6. As x increases without bound (gets larger and larger), the
graph approaches, but never reaches the horizontal axis (like
approaching an asymptote). As x decreases without bound (gets
larger and larger in the negative direction) the graph
approaches, but never reaches, the horizontal axis.
7. The Empirical Rule applies [pic] Note: we are going to use tables (for Z statistics) or our calculator not
the normal PDF!! Area under a Normal Curve
The area under the normal curve for any interval of values of
the random variable X represents either
. The proportion of the population with the characteristic
described by the interval of values or
. The probability that a randomly selected individual from the
population will have the characteristic described by the
interval of values
. [the area under the curve is either a proportion or the
probability] Standardizing a Normal Random Variable (our Z statistic from before)
X - ?
Z = ----------- where ? is the mean and
? is the standard deviation of the random variable X
? Z is normally distributed with mean of 0 and standard deviation
of 1 TI-83 Normal Distribution functions: #1: normalpdf pdf = Probability Density Function
This function returns the probability of a single value of the random
variable x. Use this to graph a normal curve. Using this function returns
the y-coordinates of the normal curve.
Syntax: normalpdf (x, mean, standard deviation) #2: normalcdf cdf = Cumulative Distribution Function
This function returns the cumulative probability from zero up to some input
value of the random variable x. Technically, it returns the percentage of
area under a continuous distribution curve from negative infinity to the
x. You can, however, set the lower bound.
Syntax: normalcdf (lower bound, upper bound, mean, standard
deviation)
#3: invNorm( inv = Inverse Normal Probability Distribution Function
This function returns the x-value given the probability region to the left
of the x-value.
(0 < area < 1 must be true.) The inverse normal probability distribution
function will find the precise value at a given percent based upon the mean
and standard deviation.
Syntax: invNorm (probability, mean, standard deviation) (the above take from
http://mathbits.com/MathBits/TISection/Statistics2/normaldistribution.htm) We can use -E99 for negative infinity and E99 for positive infinity. Example 1: A random number generator on calculators randomly generates a
number between 0 and 1. The random variable X, the number generated,
follows a uniform distribution
a. Draw a graph of this distribution
b. What is the P(0