χ²cdf( is the χ² cumulative density function. If some random variable follows a χ² distribution, you can use this command to find the probability that this variable will fall in the interval you supply.
The command takes three arguments. lower and upper define the interval in which you're interested. df specifies the degrees of freedom (choosing one of a family of χ² distributions).
Often, you want to find a "tail probability" - a special case for which the interval has no lower or no upper bound. For example, "what is the probability x is greater than 2?". The TI-83+ has no special symbol for infinity, but you can use E99 to get a very large number that will work equally well in this case (E is the decimal exponent obtained by pressing [2nd] [EE]). Use E99 for positive infinity, and -E99 for negative infinity.
The χ²cdf( command is crucial to performing a χ² goodness of fit test, which the early TI-83 series calculators do not have a command for (the χ²-Test( command performs the χ² test of independence, which is not the same thing, although the manual always just refers to it as the "χ² Test"). This test is used to test if an observed frequency distribution differs from the expected, and can be used, for example, to tell if a coin or die is fair.
As with other continuous distributions, we can define χ²cdf( in forms of the probability density function: