2.1. Overview

The MA206 program provides a single HP-48G Series program which automates many of the calculations required in the Probability and Statistics course. In particular, the MA206 program permits the user to calculate:

• Binomial interval probabilities. Calculate the probability that a random variable following a binomial distribution lies in a user specified interval.

• Poisson interval probabilities. Calculate the probability that a Poisson distributed random variable lies in a user supplied interval.

• Hypergeometric interval probabilities. Calculate the probability that a random variable from a hypergeometric distribution lies in a user supplied interval.

• Uniform interval probabilities.

• Exponential interval probabilities.

• Lower tail Normal probabilities.

• Upper tail Normal probabilities and critical values.

• Upper tail Student's T probabilities and critical values.

• Upper tail Chi-Squared probabilities and critical values.

• Clean up stored intermediate variables.

The three functions which compute upper tail probabilities and critical values use the HP-48G Series SOLVE application, with the result that any of the parameters of the distribution, the critical value, and the upper tail probability can be computed if the remaining values are known.

2.2. Program Invocation

The MA206 program is invoked by pressing the user menu key corresponding to the screen label MA206. This label will be visible when the current directory is set to the directory in which the MA206 program is installed, and user variables are selected. If user variables are not visible, pressing the VAR key will cause them to be visible.

After the MA206 key is pressed, a user selection box labeled MA206 PROGRAMS will appear, showing the first four functions. To scroll to the other functions, use the cursor arrow keys. Note that when the top of the list is visible, a "down arrow" will appear in the lower right of the box, indicating that there are more selections available below those visible. When the middle of the list is displayed, arrows will appear in both the bottom and top of the dialog box.

Even when the top or bottom of the list is displayed, you can scroll in both directions. If you scroll up off of the list, the selection moves to the bottom of the list and vice versa. This can be convenient if you know that the desired item is near the bottom of the list.

Once the desired item is highlighted, select ENTER.

2.3. Binomial

This option calculates the probability that a random variable following a binomial distribution lies in a user specified interval.

When the Binomial option is selected, the screen will display a data entry form labeled BINOMIAL DIST. X~BIN(N,P), with spaces for the distribution's parameters and the endpoints of the desired interval.

The four values to be entered are:

Once you have filled in the values appropriate to your problem, select OK and the probability will be calculated. The probability distribution, including its parameter values is displayed on stack level 3. The probability statement (including the selected interval) is shown on stack level 2, and the probability itself is shown on stack level 1.

If you enter values in the variables which do not satisfy the conditions described above, the results are not easily predictable, and will in any case be unlikely to have any practical value.

2.4. Poisson

This option allows you to calculate the probability that a Poisson distributed random variable lies in an interval you supply.

When the Poisson option is selected, the screen will display a data entry form labeled POISSON DIST. X~P(λ), λ>0, with spaces for the distribution's parameters and the (inclusive) end points of the interval. The three variables are:

Once you have filled in the values appropriate to your problem, select OK and the probability will be calculated. The probability distribution, with its parameter value, and the interval selected are displayed for your reference on levels 3 and 2 of the stack, and the probability it redisplayed on level 1 for use in further calculations.

If you enter values in the variables which do not satisfy the conditions described above, the results are not easily predictable, and will in any case be unlikely to have any practical value.

2.5. Hypergeometric

This option allows you to calculate the probability that a raandom variable from a hypergeometric distribution falls in an interval you define.

This sub-program was originally written by Major Dennis Day.

After you select the Hypergeometric option, a user input form labeled HYPERGEOMETRIC DIST. will appear. This form has spaces for the distribution's parameters and the endpoints of the desired interval:

Once you have filled in the values appropriate to your problem, select OK and the probability will be calculated. The probability distribution, including its parameter values is displayed on stack level 3. The probability statement (including the selected interval) is shown on stack level 2, and the probability itself is shown on stack level 1.

If you enter values in the variables which do not satisfy the conditions described above, the results are not easily predictable, and will in any case be unlikely to have any practical value.

2.6. Uniform

The MA206 option allows you to compute the probability that a random variable uniformly distributed on a defined interval lies in an interval you supply.

After the Uniform option is selected, a user input form labeled UNIFORM DIST. X~UNF(A,B). The input form has spaces for the distribution's parameters and the endpoints of the interval:

Once you have filled in the values appropriate to your problem, select OK and the probability will be calculated. The probability distribution, including its parameter values is displayed on stack level 3. The probability statement (including the selected interval) is shown on stack level 2, and the probability itself is shown on stack level 1.

If you enter values in the variables which do not satisfy the conditions described above, the results are not easily predictable, and will in any case be unlikely to have any practical value.

In particular, you should be cautious about the relationships between the interval [A, B] and the interval [X1, X2]. If X1 and X2 are not contained in [A, B], the returned probability will be numerically incorrect, although the program will appear to have returned correctly.

2.7. Exponential

This option allows you to compute the proabability that an exponentially distributed random variable lies in an interval you supply.

After the Exponential option is selected, a user input form labeled EXPONENTIAL DIST. X~EXP(λ). The input form has spaces for the distribution's parameters and the endpoints of the interval:

Once you have filled in the values appropriate to your problem, select OK and the probability will be calculated. The probability distribution, including its parameter values is displayed on stack level 3. The probability statement (including the selected interval) is shown on stack level 2, and the probability itself is shown on stack level 1.

If you enter values in the variables which do not satisfy the conditions described above, the results are not easily predictable, and will in any case be unlikely to have any practical value.

2.8. Normal

The Normal optional allows you to evaluate the CDF of the Normal distribution, effectively replacing the standard Normal tables.

Once you select the Normal option, a window will appear reminding you that this is the NORMAL DISTRIBUTION P(X≤x). Press the ENTER key and the user input form will appear.

This input form allows you to set the parameters of the distribution and the value of the random variable at which you want to evaluate the CDF.

Once you have filled in the values appropriate to your problem, select OK and the probability will be calculated. The probability distribution, including its parameter values is displayed on stack level 3. The probability statement (including the selected interval) is shown on stack level 2, and the probability itself is shown on stack level 1.

2.9. Using the SOLVE Form

The SOLVE application is one of the standard HP-48G Series applications. It allows the user to specify the values of all but one of the variables in an equation or expression, and solve for the remaining unknown value. (In the case of an expression, the root of the expressions is found.)

The next three of the MA206 functions use the SOLVE to let you find values associated with the upper tail probabilities of the Normal, Student's T, and Chi-Squared distributions.

Use the cursor keys to move the highlight to each known variable in turn, enter the value, and then press ENTER. The highlight will jump to the next variable; if this is not the one you want, you can again use the cursor keys to move. Once all known values are entered, more the highlight to the unknown value and press OK. After it is done calculating, the HP-48G Series will fill in the value for the unknown variable. If you then press ENTER, you will be returned to the normal stack view, with the value on level 1.

Figure 1 [TBR] shows the cursor in the first variable position, ready to enter the upper tail probability.

For additional information on using the SOLVE application, see the [HP 48 User's Guide].

2.10. Normal Find

2.11. T-Distribution

2.12. Chi-Squared

2.13. Clear Garbage

2.14. Installation

3.1. Top Level Structure

The top level structure of the program is a CHOOSE box which allows the user to select among the various functions used in MA206.

  1 <<
  2 "   MA206 PROGRAMs"
  3 { { "Binomial" 1}
  4 { "Poisson" 2}
  5 { "Hypergeometric" 3}
  6 { "Uniform" 4}
  7 { "Exponential" 5 }
  8 { "Normal" 6 }
  9 { "Normal Find" 7 }
 10 { "T-Distribution" 8 }
 11 { "Chi-Squared" 9 }
 12 { "Clear Garbage" 10 } }
 13 1 CHOOSE DROP -> c

The remainder of the program is a CASE statement which executes the various options.

3.2. The Binomial Distribution

The first choice is the binomial distribution, structured as an INFORM block to solicit the user's input, followed by a loop which calculates the binomial distribution. Finally the variables used in the calculation are PURGEed.

There is no error checking on the user input parameters, either for the distribution or for the interval.

  1 <<
  2 CASE 'c==1'
  3 THEN
  4    "BINOMIAL DIST.  X~BIN(n,p)"
  5 { { } { } 
  6   { "n:" "Enter Number of trials" 0 }
  7   { "p:" "Enter Probability" 0 }
  8   { } { }
  9   { "X1:" "Input Lower Limit" 0 }
 10   { "X2" "Input Upper Limit" 0 }
 11 } { 2 } { } { }
 12 INFORM DROP OBJ->
 13 DROP
 14 'X2' STO
 15 'X1' STO
 16 'p' STO
 17 'n' STO
 18 "X~Binomial(" 
 19 n ->STR + "," + p ->STR + ")"
 20 + '[sum](X=X1,X2,COMB(n,X)*p^X*(1-p)^(n-X))'
 21 ->NUM 'ANS' STO
 22 ANS 4 RND 
 23 ->STR
 24 "P(" X1 ->STR + "≤X≤" + X2 ->STR + ")=" 
 25 + SWAP + ANS
 26 'X1' PURGE
 27 'X2' PURGE
 28 'n' PURGE
 29 'p' PURGE
 30 'ANS' PURGE
 31 END

3.3. The Poisson Distribution

This section computes the probability that a Poisson distributed random variable lies in a user-specified interval.

The second section deals with the Poisson distribution. As with the binomial distribution, the basic calculation is a loop over the integers contained in the user supplied interval. We get the user inputs via an INFORM box, and after the computations are complete we PURGE the variables we use.

There is no error checking to establish that the endpoints of the interval are non-negative, which is required by the definition of the Poisson distribution.

  1 'c==2' THEN
  2 "POISSON DIST. X~P(λ), λ>0"
  3 { { } { } 
  4 { "λ:" "input the rate: λ >0" 0 }
  5 { } { } { } 
  6 { "x1:" "input lower Limit" 0 }
  7 { "X2:" "Input upper Limit" 0 }
  8 } { 2 } { } { }
  9 INFORM DROP OBJ-> DROP 'X2' STO
 10 'X1' STO
 11 'λ' STO
 12 "X~POISSON(λ=" λ ->STR + ")" +
 13 '[sum](X= X1,X2,EXP(-λ)*λ^X/X!)'
 14 ->NUM 'ANS' STO
 15 ANS 4 RND ->STR
 16 "P(" X1 ->STR + "≤X≤" + X2 ->STR + ")=" +
 17 SWAP + ANS 
 18 'X1' PURGE
 19 X2' PURGE
 20 'λ' PURGE
 21 'ANS' PURGE
 22 END

3.4. The Hypergeometric Distribution

The hypergeometric distribution follows the pattern of the other discrete distributions. We get the user's input via an INFORM box, loop over the relevant integers, and then PURGE the variables used.

There is no error checking on the parameters of the distribution or on the interval over which the probability is requested.

  1 'c==3' THEN
  2 "HYPERGEOMETRIC DIST."
  3 { { "M:" "number of possible successes" 0 }
  4 { "N:" "population size" 0 }
  5 { "Small 'n':" "total number of items drawn" 0 }Notice
that despite the used of the lower case "n" here, it will appear
uppercased in the INFORM box, as do all letters.
  6 { } { } { } 
  7 { "X1:" "LOWER # of desired successes" 0}
  8 { "X2:" "UPPER # of desired successes" 0} }
  9 { 2 } { } { }
 10 INFORM DROP OBJ-> DROP
 11 'X2' STO
 12 'X1' STO
 13 'n' STO
 14 'N' STO
 15 'M' STO
 16 "X~Hyp(x:" n ->STR +
 17 "," + M ->,STR + "," + 
 18 N ->STR + ")" +
 19 '[sum](X=X1,X2, COMB(M,X) *COMB(N-M,n-X)/COMB(N,n))'
 20 ->NUM 'ANS' STO ANS 4 RND ->STR "P(" X1 ->STR + "≤X≤" + X2 ->STR + ")=" + SWAP + ANS 
 21 'ANS' PURGE
 22 'X1' PURGE
 23 'X2' PURGE
 24 'n' PURGE
 25 'M' PURGE
 26 'N' PURGE
 27 END

3.5. The Uniform Distribution

This section computes the probability that a uniformly distributed random variable lies in a specific interval.

The uniform distribution is the first of the continuous distributions dealt with in this program. The structure of the program is similar to the preceding discrete sections, except that in the computation section we are able to compute the desired interval probability directly in a single expression.

There appears to be no error checking to ensure that the interval over which the probability is desired is contained in the interval over which the probability is non-zero, although the computational formula employed assumes that this is so.

  1 'c==4' THEN
  2 "UNIFORM DIST. X~UNF(a,b)"
  3 { { } { } 
  4 { "A:" "Enter A" 0 }
  5 { "B:" "Enter B" 0 }
  6 { } { }
  7 { "X1:" "Enter Lower limit" 0 }
  8 { "X2:" "Enter upper limit" 0 }
  9 } { 2 } { } { } 
 10 INFORM DROP OBJ-> DROP
 11 'X2' STO
 12 'X1' STO
 13 'B' STO
 14 'A' STO
 15 "X~Uniform(x;" A ->STR +
 16 "," + B ->STR + ")" +
 17 '[integral](X1, X2, 1/(B-A), X)'
 18 ->NUM 'ANS' STO
 19 "P(" X1 ->STR + "≤X≤" + X2 ->STR + ")=" + ANS 4 RND ->STR + ANS 
 20 'X1' PURGE
 21 'X2' PURGE
 22 'A' PURGE
 23 'B' PURGE
 24 'ANS' PURGE
 25 'IERR' PURGE
 26 END

3.6. The Exponential Distribution

Similar in concept to the uniform distribution, this subprogram uses an INFORM box to get the user's input, stores the results in named variables, computes the relevant probability, and PURGEs the variables before it exits.

In this case, the computational section evaluates the probability in question by directly integrating the PDF over the user supplied interval.

There is no error checking on the parameter of the distribution (which must be non-negative), or the user-supplied interval.

  1 'c==5' THEN
  2 "Exponential DIST.x~exp(λ)"
  3 { { } { }
  4 { "λ:" "Enter the rate: λ>0" 0 }
  5 { } { } { } 
  6 { "X1:" "Enter the LOWER limit" 0 }
  7 { "X2:" "Enter the UPPER limit" 0 }
  8 } { 2 } { } { }
  9 INFORM DROP OBJ->
 10 DROP 
 11 'X2' STO
 12 'X1' STO
 13 'λ' STO
 14 '[integral](X1, X2, λ *EXP(-λ*X), X)'
 15 ->NUM
 16 'ANS' STO
 17 "X~EXP(x:" λ ->STR + ")" + "P(" X1 ->STR 
 18 + "≤X≤" + X2 ->STR + ")=" + ANS 4 RND->STR + ANS 
 19 'X1' PURGE
 20 'X2' PURGE
 21 'λ' PURGE
 22 'ANS' PURGE
 23 END

3.7. Lower Tail Normal Probabilities

This subprogram finds the lower tail probabilities associated with the Normal distribution. In other words, it evaluates the Normal CDF.

The general form of the subprogram is similar to the majority of the other subprograms; an INFORM dialog is used to obtain the user's input, the relevant values are stored in named variables, the computation is performed, and the stored variables are cleared.

The computational section uses the built-in HP-48G Series function UTPN (Upper Tail Probability Normal).

The program performs no error checking.

  1 'c==6' THEN
  2 "     NORMAL      DISTRIBUTION       P(X≤x)"
  3 MSGBOX CLEAR 
  4 "Normal (lower tail) X~n(µ,σ)"
  5 { { } { }
  6 { "x:" "Enter X" 0 }
  7 { } { }
  8 { "µ:" "input the mean" 0 }
  9 { "σ:" "input the Standard Deviation" 0 }
 10 } { 2 } { } { }
 11 INFORM DROP OBJ-> DROP
 12 'σ' STO
 13 'µ' STO
 14 'σ^2' EVAL 
 15 x UTPN 'ANS' STO
 16 "X~Normal(x;"
 17 µ->STR + "," + σ ->STR + ")" 
 18 + "P(X≤" x ->STR + ")=" + '1-ANS' ->NUM 4 RND ->STR 
 19 + '1-ANS' EVAL
 20 'x' PURGE
 21 'µ' PURGE
 22 'σ' PURGE
 23 'ANS' PURGE
 24 END

3.8. Upper Tail Normal Probabilities

This program allows the user to compute values associates with the upper tail (survival) probabilities of the Normal distribution.

In contract to the preceding subprograms, this program defines an appropriate equation and then invokes the HP-48G Series SOLVE application. The user interaction is delegated to the SOLVE application, as is the display of the answer (and, for the matter, the choice of which answer the user is seeking).

  1 'c==7' THEN 
  2 "     NORMAL     DISTRIBUTION    (Upper Tail)" MSGBOX
  3 <<
  4 UPPER µ σ2 X UTPN -
  5 >>
  6 STEQ # 737281d LIBEVAL
  7 END

3.9. Upper Tail T Probabilities

This subprogram allows the user to compute values associated with the upper tail of Student's T distribution.

As with the upper tail Normal subprogram, this subprogram invokes the SOLVE application to obtain user input and perform the relevant calculations.

  1 'c==8' THEN
  2 "t-DISTRIBUTION
  3  (Upper Tail)" MSGBOX
  4 <<
  5 UPPER df T UTPT -
  6 >>
  7 STEQ
  8 # 737281d LIBEVAL END

3.10. Upper Tail Chi-Squared Probabilities

This subprogram allows the user to obtain values associated with the upper tail of the Chi-Squared distribution.

Similar in implementation to the other upper tail subprograms, this section uses the HP-48G Series SOLVE application.

  1 'c==9' THEN
  2 "    Chi-Squared    DISTRIBUTION 
  3 (Upper Tail)" MSGBOX
  4 <<UPPER df X2 UTPC -
  5 >> STEQ
  6 # 737281d LIBEVAL END

3.11. Cleanup

This subprogram PURGEs the variables which may be left in the user's directory by the other subprograms.

No error checking is done before purging these variables. Indeed, it is difficult to see how error checking could be performed, given the overall structure of the MA206 program. This issue is further discussed below, as it is a general design issue for the program, not just this subprogram.

  1 'c==10' THEN
  2 'EXPR' PURGE
  3 'EQ' PURGE
  4 'UPPER' PURGE
  5 'df' PURGE
  6 'T' PURGE'
  7 'X2' PURGE
  8 'µ' PURGE
  9 'σ2' PURGE
 10 'X' PURGE
 11 'X1' PURGE
 12 'n' PURGE
 13 'p' PURGE
 14 'ANS' PURGE
 15 'λ' PURGE
 16 'M' PURGE
 17 'N' PURGE
 18 'A' PURGE
 19 'B' PURGE
 20 'λ' PURGE This second PURGE
of λ appears to be redundant.
 21 'x' PURGE
 22 'σ' PURGE
 23 'IERR' PURGE
 24 END
 25 END
 26 >>
 27 >>

4.1. Overwriting Stored Variables

A consequence of this is that if any of these variables are used other than as temporary variables in the MA206 program, the MA206 program may overwrite or delete them without warning. Since a number of the variables were chosen to have mnemonic value matching the names of variables commonly used in the subject area.... [TBR]

4.2. Calling the SOLVE Application

Neat Trick! [TBR]

4.3. Using the INFORM Dialog

As it is used in this program, the INFORM programming construct causes a program error if the user selects CANCEL from within the INFORM screen. This occurs because when the user exits INFORM, the routine returns a status value, which is DROPped without examination in the existing program, and the resulting stack does not contain the right number or type of values.

Causing the program to exit with an error is not catastrophic. The error occurs if the user selects CANCEL, and results in program termination. Since the user presumably wants out of the program if they select CANCEL, the actual result matches reasonably with the user's expectation. However, creating a program error (and thus displaying an apparently unrelated error message) is bad programming style because it may be confusing, especially to an unsophisticated user.

5.1. Normal Intervals