## Continuous Probability Distributions

A continuous probability distribution is a representation of a variable that can take a continuous range of values.

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### Key Takeaways

Key PointsA probability density function is a function that describes the relative likelihood for a random variable to take on a given value.Intuitively, a continuous random variable is the one which can take a continuous range of values — as opposed to a discrete distribution, where the set of possible values for the random variable is at most countable.While for a discrete distribution an event with probability zero is impossible (e.g. rolling 3 and a half on a standard die is impossible, and has probability zero), this is not so in the case of a continuous random variable.Key Terms**Lebesgue measure**: The unique complete translation-invariant measure for the

A continuous probability distribution is a probability distribution that has a probability density function. Mathematicians also call such a distribution “absolutely continuous,” since its cumulative distribution function is absolutely continuous with respect to the Lebesgue measure

Intuitively, a continuous random variable is the one which can take a continuous range of values—as opposed to a discrete distribution, in which the set of possible values for the random variable is at most countable. While for a discrete distribution an event with probability zero is impossible (e.g. rolling 3 and a half on a standard die is impossible, and has probability zero), this is not so in the case of a continuous random variable.

For example, if one measures the width of an oak leaf, the result of 3.5 cm is possible; however, it has probability zero because there are uncountably many other potential values even between 3 cm and 4 cm. Each of these individual outcomes has probability zero, yet the probability that the outcome will fall into the interval (3 cm, 4 cm) is nonzero. This apparent paradox is resolved given that the probability that

The definition states that a continuous probability distribution must possess a density; or equivalently, its cumulative distribution function be absolutely continuous. This requirement is stronger than simple continuity of the cumulative distribution function, and there is a special class of distributions—singular distributions, which are neither continuous nor discrete nor a mixture of those. An example is given by the *Cantor distribution*. Such singular distributions, however, are never encountered in practice.

### Probability Density Functions

In theory, a probability density function is a function that describes the relative likelihood for a random variable to take on a given value. The probability for the random variable to fall within a particular region is given by the integral of this variable’s density over the region. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.

Unlike a probability, a probability density function can take on values greater than one. For example, the uniform distribution on the interval

### Key Takeaways

Key PointsThe distribution is often abbreviated**cumulative distribution function**: The probability that a real-valued random variable

**p-value**: The probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true.

**Box–Muller transformation**: A pseudo-random number sampling method for generating pairs of independent, standard, normally distributed (zero expectation, unit variance) random numbers, given a source of uniformly distributed random numbers.

The continuous uniform distribution, or rectangular distribution, is a family of symmetric probability distributions such that for each member of the family all intervals of the same length on the distribution’s support are equally probable. The support is defined by the two parameters,

The probability that a uniformly distributed random variable falls within any interval of fixed length is independent of the location of the interval itself (but it is dependent on the interval size), so long as the interval is contained in the distribution’s support.

To see this, if

Is independent of

### Applications of the Uniform Distribution

When a

### Sampling from a Uniform Distribution

There are many applications in which it is useful to run simulation experiments. Many programming languages have the ability to generate pseudo-random numbers which are effectively distributed according to the uniform distribution.

If

### Sampling from an Arbitrary Distribution

The uniform distribution is useful for sampling from arbitrary distributions. A general method is the inverse transform sampling method, which uses the cumulative distribution function (CDF) of the target random variable. This method is very useful in theoretical work. Since simulations using this method require inverting the CDF of the target variable, alternative methods have been devised for the cases where the CDF is not known in closed form. One such method is rejection sampling.

The normal distribution is an important example where the inverse transform method is not efficient. However, there is an exact method, the Box–Muller transformation, which uses the inverse transform to convert two independent uniform random variables into two independent normally distributed random variables.

### Example

Imagine that the amount of time, in minutes, that a person must wait for a bus is uniformly distributed between 0 and 15 minutes. What is the probability that a person waits fewer than 12.5 minutes?

Let

We want to find **Erlang distribution**: The distribution of the sum of several independent exponentially distributed variables.**Poisson process**: A stochastic process in which events occur continuously and independently of one another.

### Key Takeaways

Key PointsThe mean of a normal distribution determines the height of a bell curve.The standard deviation of a normal distribution determines the width or spread of a bell curve.The larger the standard deviation, the wider the graph.Percentiles represent the area under the normal curve, increasing from left to right.Key Terms**empirical rule**: That a normal distribution has 68% of its observations within one standard deviation of the mean, 95% within two, and 99.7% within three.

**bell curve**: In mathematics, the bell-shaped curve that is typical of the normal distribution.

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**real number**: An element of the set of real numbers; the set of real numbers include the rational numbers and the irrational numbers, but not all complex numbers.