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Last Update
5th May 2011
SESSION 37 & 38
Continuous Probability Distributions
Lecturer:
University:
Domain:
Florian Boehlandt
University of Stellenbosch Business School
http://www.hedge-fundanalysis.net/pages/vega.php
Learning Objectives
1. Normal Distribution
2. Z-Score and Transformation
Quantitative Data
Data variables
can only assume
certain values
and are collected
typically by
counting
observations
Data
Discrete
Continuous
Binomial and
Poisson
distribution
Continuous
probability
distribution
functions (most
notably normal,
student-t, F,
Chi-Squared)
Data variables
can assume any
value within a
range (real
numbers) and
are collected by
measurement.
Normal Distribution
The normal distribution is the most important of all continuous
distribution functions due to its importance in statistical
inference. The probability density function of a normal random
variable is:
Thus, for every x the function is characterized by the population
mean μ and population variance σ. The normal distribution
function is symmetric about the mean and the random variable
ranges between -∞ and +∞. The function is asymptotic.
Normal Probabilities
To calculate the probability that a normal random variable falls
into any interval, the area in the interval under the curve needs
to be computed. This requires calculations that are beyond the
scope of introductory integral calculus. Fortunately, probability
tables may be used to determine the area underneath the curve.
This requires the use of a standard normal variable:
Assuming the population parameters are known, Z may be
calculated for any X. This is called Z-score conversion. The
corresponding probabilities (area underneath the curve) are
then determined from normal probability tables.
Normal Probabilities Table
THE STANDARD NORMAL DISTRIBUTION (Z) – Areas under the curve
Z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
…
…
…
…
…
…
…
…
…
…
…
1.0
0.3413
0.3438
0.3481
0.3485
0.3508
0.3531
0.3554
0.3577
0.3599
0.3621
1.1
0.3643
0.3665
0.3686
0.3708
0.3729
0.3749
0.3770
0.3790
0.3810
0.3830
1.2
0.3849
0.3869
0.3888
0.3907
0.3925
0.3944
0.3982
0.3980
0.3997
0.4015
1.3
0.4032
0.4049
0.4066
0.4082
0.4099
0.4115
0.4131
0.4147
0.4162
0.4177
1.4
0.4192
0.4207
0.4222
0.4236
0.4251
0.4265
0.4279
0.4292
0.4306
0.4319
1.5
0.4332
0.4345
0.4357
0.4370
0.4382
0.4394
0.4406
0.4418
0.4429
0.4441
The number s in the left column describe the values of z to one decimal
place and the column headings specify the second decimal place. The
table provides the probability that a standard normal variable falls between
0 and values of z. It is important to note that the normal curve is symmetric
as only one half of the distribution function is displayed in most textbooks
(e.g. P(Z > 0) = P(Z < 0) = 0.5). It is assumed that P(Z > 3.1) ≈ 0.
Example
Suppose that the amount of time to assemble a computer
(assembly time is an interval variable) is normally distributed
with a mean μ = 50 minutes and a standard deviation σ = 10
minutes. What is the probability that the computer is assembled
in a time between 45 and 60 minutes? (i.e. we want to find the
probability P(45 < X < 60)).
Solution
Step 1: Transformation of the variables
Step 2: Determine the probabilities for the Z-scores
The probability we seek is actually the sum of two probabilities:
Or
Exercise
Consider an investment whose return is normally distributed
with a mean of 10% and a standard deviation of 5%.
a) What is the probability of loosing money (Hint: P(X < 0))?
b) Find the probability of loosing money when the standard
deviation is equal to 10%.
Finding values of z
There is a family of problems that require researchers to
determine the value of Z given a certain probability. From the
financial area, for example, value-at-risk:
What is the rate of return that 5% of all observations fall below
of?
This requires reading of a probability from the normal probability
tables and determine the associated z-score. From there, we can
use a re-arranged version of the previous formula to determine
X:
Central Limit Theorem
The sampling distribution of the mean of a random sample
drawn from any population is approximately normal for
sufficiently large sample sizes. The larger the sample size, the
more closely the sampling distribution of x-bar will resemble the
normal distribution.