$ a $ | $ = $ | $ b $ | $ = $ |

$P(x$ $)$

$P($ $x$ $)$

$E(x)$

$\text{Var}(x)$

$P(z$ $)$

$P($ $x$ $)$

$\text{Area to the }$$\text{ of } z \text{ is } $

$\text{Area }$ $\text{and}$ $\text{is}$

$ \mu $ | $ = $ | $ \sigma $ | $ = $ |

$P(x$ $)$ |

$P($ $x$ $)$

$x \text{ is in the }$ %

$ n $ | $ = $ | $ p $ | $ = $ |

$\mu$

$\sigma$

$P(x$ )

$ \mu $ | $ = $ |

$P(x$ $)$ |

$P($ $x$ $)$

$ df $ | $ = $ |

$\text{ of } t=$

$P($ $x$ $)$

$\text{ tail area is }$

$\text{Area }$ $\text{and}$ $\text{is}$

$ df $ | $ = $ |

$\text{Upper tail area is }$

$ \text{Num. } df $ | $ = $ | $ \text{Den. } df $ | $ = $ |

$\text{Upper tail area is }$

Example 1 • Example 2

Example 1 • Example 2

Example 1 • Example 2

Example 1 • Example 2

Example 1 • Example 2

Example 1 • Example 2

Example 1 • Example 2

Example 1 • Example 2

How it Works:

For a continuous probability distribution, probability is calculated by taking the area under the graph of the probability density function, written f(x). For the uniform probability distribution, the probability density function is given by f(x)=$\begin{cases} \frac{1}{b-a} \quad \text{for } a \leq x \leq b \\ 0 \qquad \, \text{elsewhere} \end{cases}$. The graph of this function is simply a rectangle, as shown below. Then the area under the graph of f(x) over some interval is also going to be a rectangle, which can easily be calculated as length$\times$width. Furthermore, the expected value and variance for a uniformly distributed random variable are given by E(x)=$\frac{a+b}{2}$ and Var(x) = $\frac{(b-a)^2}{12}$, respectively.

The most important continuous probability distribution is the normal probability distribution. Its graph is bell-shaped and is defined by its mean ($\mu$) and standard deviation ($\sigma$). The mean is the highest point on the curve and the standard deviation determines how flat the curve is. Obviously, this is a much more complicated shape than the uniform probability distribution. The area under it can't be calculated with a simple formula like length$\times$width. So, instead, we rely on the standard normal probability distribution to calculate probabilities for the normal probability distribution.

The standard normal probability distribution (or z distribution) is simply a normal probability distribution with a mean of 0 and a standard deviation of 1. There are three types of probabilities to know how to compute for the z distribution: (1) the probability that z will be less than or equal to a value, (2) the probability that z will be between two values and (3) the probability that z will be greater than or equal to a value. We can find these probabilities using the standard normal table (or z-table).

Standard Normal Table | |||||

z | .00 | .01 | .02 | .03 | .04 |

0.0 | .50000 | .50399 | .50798 | .51197 | .51595 |

0.1 | .53983 | .54380 | .54776 | .55172 | .55567 |

0.2 | .57926 | .58317 | .58706 | .59095 | .59483 |

Now that we know how to calculate probabilities for the z-distribution, we can calculate probabilities for any normal distribution. We can do this by converting from normal to standard normal, using the formula $z=\frac{x-\mu}{\sigma}$. So, given a problem to calculate probability for a normal distribution, we start by converting the values to z-values. Then, depending on the type of z distribution probability type it is, we rewrite the problem so it's in terms of the probability that z less than or equal to a value. Then we use the z-table to find those probabilities and compute our answer.