SHAZAM Computing Probabilities

Computing Probabilities for Normal Random Variables


The cumulative distribution function (CDF) for the standard normal random variable can be computed with the NCDF function on the GENR command. The command format is:

GENR prob=NCDF(var)

where var is a variable that contains numbers and prob is a variable that will contain the probabilities.

If the probability for only one number is required then the GEN1 command should be used. The command format is:

GEN1 prob=NCDF(z)

where z is a number or scalar variable.


The DISTRIB command can be used for the calculation of the properties of a variety of probability distributions. For the normal distribution, the general format of the DISTRIB command is:

DISTRIB var / TYPE=NORMAL options

where var is a variable with values and options is a list of desired options. The option TYPE=NORMAL specifies that the normal distribution is required. Some useful options are:

INVERSE Computes critical values of the distribution. The data in the var variable must be entered in probabilities.
MEAN= Specifies the value of the population mean. For the normal distribution the default is MEAN=0.
VAR= Specifies the value of the population variance. For the normal distribution the default is VAR=1.

Options that save results
CDF= Saves the values of the cumulative distribution function in the variable specified.
CRITICAL= Saves the critical values in the variable specified when the INVERSE option is used.

Example

This example is from an exercise in Newbold [1995, Exercise 19, Chapter 5]. Suppose that anticipated consumer demand for a product next month can be represented by a normal random variable with mean 1200 units and standard deviation 100 units.

Denote the random variable by X. For some value x, the CDF is P(X<x)=F(x). The random variable Z=(X-1200)/100 has a standard normal distribution (mean 0 and variance 1).

  Questions and Solutions

(a) What is the probability that sales will exceed 1000 units ? We need to find:

         P(X > 1000) = 1 - P(X < 1000)
                     = 1 - F(1000)

Note that:

         P(X < 1000) = P(Z < (1000-1200/100) )
                     = P(Z < -2)

(b) What is the probability that sales will be between 1100 and 1300 units ? We need to find:

         P(1100 < X < 1300) = P(X < 1300) - P(X < 1100)
                            = F(1300) - F(1100)

Note that:

         P(X < 1100) = P(Z < (1100-1200/100) ) = P(Z < -1)     and

         P(X < 1300) = P(Z < (1300-1200/100) ) = P(Z < 1) 

(c) The probability is 0.10 that sales will be more than how many units ? We need to find a value b such that:

         P(X > b) = .10

The SHAZAM commands (filename: NPROB.SHA) that follow show how to compute the answers. Note that the GEN1 command is used for scalar arithmetic.

SAMPLE 1 1 
* Part (a)
GEN1 ANSWER = 1 - NCDF((1000-1200)/100)
PRINT ANSWER

* Part (b)
GEN1 ANSWER = NCDF((1300-1200)/100) - NCDF((1100-1200)/100)
PRINT ANSWER

* Part (c)
* Compute the variance of X
GEN1 SIG2=100**2
GEN1 PROB=.1
DISTRIB PROB / TYPE=NORMAL MEAN=1200 VAR=SIG2 INVERSE CRITICAL=ANSWER
PRINT ANSWER
STOP

The SHAZAM output can be viewed. The answers to the questions are:

(a)   P(X > 1000) = 0.977

(b)   P(1100 < X < 1300) = 0.683

(c)   P(X > 1328) = .10


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SHAZAM output


 |_SAMPLE 1 1
 |_* Part (a)
 |_GEN1 ANSWER = 1 - NCDF((1000-1200)/100)
 |_PRINT ANSWER
     ANSWER
    .9772499

 |_* Part (b)
 |_GEN1 ANSWER = NCDF((1300-1200)/100) - NCDF((1100-1200)/100)
 |_PRINT ANSWER
     ANSWER
    .6826895

 |_* Part (c)
 |_* Compute the variance of X
 |_GEN1 SIG2=100**2
 |_GEN1 PROB=.1
 |_DISTRIB PROB / TYPE=NORMAL MEAN=1200 VAR=SIG2 INVERSE CRITICAL=ANSWER
 NORMAL DISTRIBUTION - MEAN=   1200.0     VARIANCE=   10000.
 
               PROBABILITY CRITICAL VALUE   PDF
   PROB
  ROW     1     .10000      1328.2      .17550E-02
 |_PRINT ANSWER
     ANSWER
    1328.155
 |_STOP

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