Statistics Definitions > Probability Density Function

Continuous variables can take on an infinite number of variables. For example, you could have a random variable Y that represents possible weights:

The probability density function(also called a probability distribution function) shown above shows all possible values for Y, which for this case, has an infinite amount of possibilities. For example, the random variable Y could equal 180 pounds, 151.2 pounds or 201.9999999999 pounds.

We can use probability density functions to answer a question like:

**What is the probability that a person will weigh between 150 lbs and 250 lbs? **

Written in notation, the question becomes:

P(150 < Y < 250)

To answer the question, shade the area on the graph:

Then approximate the area. From looking at the shaded area, it looks like it’s about 75 percent.

So, P(150 < Y < 250) = 75%.

You can use the same technique for figuring out the probability for less than or greater than a certain number. Just shade in the area to the right of the number (for greater than) or to the left of the number (for less than).

### Use Caution When Reading Probability Density Function Graphs!

Probability functions are great for figuring out intervals (because then you have an area to measure). However, you have to use a little caution with reading probability density function graphs, especially when it comes to exact numbers. For example: What about the probability any person will weigh **exactly **180lbs? Written in notation, the question would be:

P(Y=180).

Looking at the graph, you might think that the probability of a person weighing 180lbs is about 50%, But that doesn’t make sense, right? That *half* of all people weigh exactly 180lbs! What you have to think about is that someone could weigh 180 pounds or they could weigh 180 pounds and a fraction of an ounce either way. they could weigh 180.00001 pounds or they could weight 179.999999999 pounds. In fact, the odds of someone weighing **exactly **180 pounds is so tiny it’s practically zero.

Another way to look at this is that if you drew the “area” for a question like this, it would actually just be a line. And a line has zero area!

**Note**: We used a normal distribution in the above example, but probability density functions can be any shape, including uniform distributions and exponential distributions.

## TI 83 NormalPDF Function

The **TI 83 normalPDF** function, accessible from the DISTR menu will calculate the normal probability density function, given the mean μ and standard deviation σ. The function doesn’t actually give you a probability, because the normal distribution curve is continuous. However, you can use it to plot a bell curve and to find x-values and y-values for points on the curve.

For other normal distribution commands on the TI 83 (like the TI 83 normalCDF), visit the TI 83 for statistics menu.

## TI 83 NormalPDF function: Steps

Watch the video or read the steps below:

**Sample Problem**: Graph a bell curve on the **TI 83 calculator** with a mean of 100 and standard deviation of 15. Use the NormalPDF function.

**Step 1:** Press Y=.

**Step 2:** Press 2nd VARS 1 to get “normalPDF.”

**Step 3:** Press the **X,T,θ,n** button, then the mean (100), then the standard deviation, 15. Close the parentheses.

**Step 4:** Press WINDOW.

**Step 5:** Change the window values to the following (type the values into the relative boxes):

Xmin=100-3*15

Xmax=100+3*15

Xscl=1

Ymin=0

Ymax=normalpdf(100,100,15)

Notes:

- The x-min/x-max is set to the mean, minus/plus three standard deviations, as this is a bell curve so +/- 3 standard deviations will show the entire curve.
- Ymax uses the normalpdf function to determine the maximum y-value at the mean (the peak of the curve)

**Step 6:** Press GRAPH. The TI 83 will graph a normal distribution curve on your screen.

**Step 7:** Press TRACE and then type in any number to find the y-value. For this example, type 80 and then press ENTER.

That’s how to use the TI 83 NormalPDF!

**Tip:** A mean of zero and a standard deviation of are the default values for a normal distribution on the calculator, if you don’t set those values.

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