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

$P(x$ $)$

$P($ $x$ $)$

$E(x)$

$\text{Var}(x)$

$P(z$ $)$

$P($ $x$ $)$

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

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

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

$P(x$ $)$ |

$P($ $x$ $)$

$x \text{ is in the }$ %

$n = $ | $p = $ |

$\text{Expected Value}$

$\text{Standard Deviation}$

$P(x$ )

$ \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{Numerator } df$ | $= $ | |

$ \text{Denominator } 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

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), a portion of which is shown below.

Standard Normal Table | |||||

${\color{Black} z}$ | .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.

The normal probability distribution can be used to approximate probabilities for the binomial probability distribution. This may be necessary in situations where the binomial probabilities are difficult to compute. This calculation is done using the continuity correction factor. This is necessary because the normal distribution is a continuous distribution while the binomial distribution is a discrete distribution. Since the probability of a single value is zero in a continuous distribution, adding and subtracting .5 from the value and finding the probability in between solves this problem.

The exponential probability distribution is useful in describing the time and distance between events. The probability density function for an exponential distribution is given by $ f(x) = \frac{1}{\mu} e^{-x/\mu}$ for x>0. Probabilities for the exponential distribution are not found using the table as in the normal distribution. They involve using a formula, although a more complicated one than used in the uniform distribution. The formula for calculating probabilities in an exponential distribution is $ P(x \leq x_0) = 1 - e^{-x_0/\mu} $.

The t-distribution is similar to the standard normal distribution. They both have a similar bell-shape and finding probabilities involve the use of a table. The main difference is that the t-distribution depends on the degrees of freedom. We have a different t-distribution for each of the degrees of freedom. Another difference is that the t table provides the area in the upper tail whereas the z table provides the area in the lower tail. So use of the t table involves matching the degrees of freedom with the area in the upper tail to get the corresponding t-value.

t-Distribution Table | ||||||

Area in the Upper Tail | ||||||

df | .20 | .10 | .05 | .025 | .01 | .005 |

1 | 1.376 | 3.078 | 6.314 | 12.706 | 31.821 | 63.656 |

2 | 1.061 | 1.886 | 2.920 | 4.303 | 6.965 | 9.925 |

3 | .978 | 1.638 | 2.353 | 3.182 | 4.541 | 5.841 |

Continuous probability distributions are probability distributions for continuous random variables. A closely related topic in statistics is discrete probability distributions. Discrete distributions are probability distributions for discrete random variables. Probabilities for discrete probability distributions can be found using the Discrete Distribution Calculator. The most important continuous probability distributions is the normal probability distribution. It is used extensively in statistical inference, such as sampling distributions. Sampling distributions can be solved using the Sampling Distribution Calculator.