07 Nov Solving a probability density function of the standard normal distribution with the HP15C/DM15L

The probability density function (PDF) of the standard normal distribution, often denoted as the bell-shaped curve or Gaussian distribution, serves as a fundamental tool in statistics and probability theory. It quantifies the likelihood of a random variable taking on a specific value, with higher probabilities associated with values near the mean and lower probabilities for values further from the mean. This PDF is invaluable for various applications, such as hypothesis testing, confidence intervals, and estimating the likelihood of events occurring within a continuous range.

If you are as ancient as I am, you will remember that probabilities were read of a t or z table. But there is no fun in that, is there? I was quite surprised that the HP10BII+ had this functionality already programmed in. Not even the venerable HP12C can calculate the P value. So well done for the HP10BII+. In any case, I had to find a way to calculate it with my DM15L/HP15C by integrating the formula.

So the basic formula is as follows:

\(\Large \tag{1} f(x)= \frac {e^{-0.5x^2}}{(\sqrt{2\pi})^2}\)

In order to find the probability interval, we need to integrate the equation starting from the lower value to upper value, for instance this could be from -1 to 1, which should be about 68.27% or 0.6827. The table might give a slightly different value.

To find the probability between -1 and 1, we integrate the following equation:

\(\Large \tag{2}\displaystyle \int_{-1}^1\frac {e^{-0.5x^2}}{(\sqrt{2\pi})^2}dx\)

Programming it into the DM15L/HP15C is easy:

[f][R↓] 
[g][R/S] 
[f][SST]C 
[√x]
[0.5][CHS]
[x]
[ex]
[2]
[ENTER]
[g][EEX]
[X] 
[g][GSB] 
[g][R/S]

With the program saved to label C, we can start the integration process:

[1][CHS]
[ENTER]
[1] (do not press ENTER)
[f]
[x]
[10x] (The program is stored in label C)

Wait a couple of seconds and the answer should be 0.6827.