What is Continuous Probability Distribution?

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

A continuous distribution defines the possible values of a continuous random variable. A continuous random variable has an infinite and uncountable set of possible values (known as the range).

Continuous Variable in the Continuous Probability Distribution

If the set of possible values of a random variable or random continuous variable X is either a single interval on the number line or a union of disjoint intervals, the variable is said to be continuous.

Discrete vs Continuous Distribution

A discrete probability distribution is one that accepts only a finite number of values such as integer values. A continuous distribution is one in which data can take on any value within a given range of values (which can be infinite).

For a discrete probability distribution, the values in the distribution will be given with probabilities.

"The probability that the web page will receive 12 clicks in an hour is 0.15," for example.

A continuous distribution, on the other hand, has an infinite number of potential values, and the probability associated with any one of those values is null. As a result, probability density is often used to define continuous distributions, which can be translated into the likelihood of a value falling within a given range.

Types of Continuous Probability Distribution

The following are the most common continuous probability distributions.

1. Normal Distribution

This is the most widely debated and encountered distribution in the real world. Given a large enough sample, several continuous distributions can converge to a normal distribution.

The mean and standard deviation are the two parameters for the normal distribution.

This distribution has a number of fascinating characteristics. The mean has the highest probability, and all other values are symmetrically distributed to either side of the mean. The standard normal distribution is a subset of the distribution with a mean of 0 and a standard deviation of 1.

This also follows an empirical formula in which 68% of the values are one standard deviation away from the mean, 95% are two standard deviations away, and 99.7% are three standard deviations away. This property comes in handy when creating hypothesis checks.

The probability density function (PDF) is given by,

f (x) = \frac{1}{\sqrt{2 π} σ} \exp (- \frac{1}{2} {(\frac{x - μ}{σ})}^{2})

Where μ is the mean of the random variable X and σ is the standard deviation.

2. Continuous Uniform Distributions

There are two types of uniform distribution: continuous and discrete. We'll talk about the continuous one here. The random variables whose values have equal chances of occurring are plotted in this distribution. Flipping a fair die is the most common example. All six outcomes are equally likely to occur in this case. As a result, the likelihood remains constant.

Consider the following distribution with a = 10 and b = 20, which looks like this:

The PDF is given by,

f (x) = \{\begin{cases} \frac{1}{b - a}, x \in [a, b] \\ 0, otherwise \end{cases}

Where a denotes the lowest value and b denotes the highest value.

3. Log-Normal Distribution

The random variables whose logarithm values obey a log-normal Distribution are plotted using this distribution. Take a look at the random variables X and Y. The variable defined in this distribution is Y = ln(X), whereby ln represents the natural logarithm of X values.

The PDF is given by,

f (x) = \frac{1}{x \sqrt{2 π} σ} \exp (- \frac{1}{2} {(\frac{\ln x - μ}{σ})}^{2})

Where μ is the mean of Y and σ is the standard deviation of Y.

4. Exponential Distribution

The exponential distribution (also referred to as the negative exponential distribution) describes the time between the two events.

The Poisson distribution and the exponential distribution have a close relationship. Let's assume a Poisson distribution is used to model the number of births over a given time span. Then an exponential distribution can be used to model the time between each birth.

The PDF is given by,

f (x) = λ e^{- λ x}

Where λ is the rate parameter and λ = 1/ (average time between events).

5. Chi-Square Distribution

The sum of squares of p normal random variables yield this distribution function. The number of degrees of freedom is denoted by p. As with the t-distribution, the distribution steadily approaches the normal distribution as the degrees of freedom increase. A three-degree-of-freedom chi-square distribution is shown below.

The PDF is given by,

f (x) = \frac{x^{\frac{p}{2} - 1} e^{- \frac{x}{2}}}{2^{\frac{p}{2}} Γ (\frac{p}{2})}

Where p denotes the number of degrees of freedom and Γ denotes the gamma function.

The following formula is used to measure the chi-square value:

χ^{2} = \sum_{i} \frac{{(O_{i} - E_{i})}^{2}}{E_{i}}

Where O denotes the actual value and E denotes the predicted value. This is used to draw inferences about the population variance of normal distributions in hypothesis testing.

6. Student’s T Distribution

The t distribution for a student is said to be identical to that of normal distribution. The difference is distribution's tails are smaller. Where the sample size is small and the population variance is unknown, this method is used. The degree of freedom (p) of this distribution function is determined as one less than the sample size, i.e., (n – 1).

The t-distribution approaches the normal distribution as the sample size and degree of freedom increases and then the tails get narrower, and the curve gets closer to the value of the mean. Where the sample size is less than 30 and the population variance is uncertain, this distribution is used to evaluate estimates of the population mean. The t-value is calculated using the sample variance/standard deviation.

The PDF is given by,

f (t) = \frac{Γ (\frac{p + 1}{2})}{\sqrt{p π} Γ (\frac{p}{2})} {(1 + \frac{t^{2}}{p})}^{- (\frac{p + 1}{2})}

Where p denotes the number of degrees of freedom and Γ denotes the gamma function.

In hypothesis testing, the t-statistic is determined as follows:

t = \frac{\bar{x} - μ}{\frac{s}{\sqrt{n}}}

Where x̄ represents the sample mean, μ represents the population mean, and s represents the sample variance.

Formulas

The PDF of a normal distribution is given by,

f (x) = \frac{1}{\sqrt{2 π} σ} \exp (- \frac{1}{2} {(\frac{x - μ}{σ})}^{2})

Where μ is the mean of the random variable X and σ is the standard deviation.

The PDF of a continuous uniform distribution is given by,

f (x) = \{\begin{cases} \frac{1}{b - a}, x \in [a, b] \\ 0, otherwise \end{cases}

Where a denotes the lowest value and b denotes the highest value.

The PDF of a log-normal distribution is given by,

f (x) = \frac{1}{x \sqrt{2 π} σ} \exp (- \frac{1}{2} {(\frac{\ln x - μ}{σ})}^{2})

Where μ is the mean of Y and σ is the standard deviation of Y.

The PDF of an exponential distribution is given by,

f (x) = λ e^{- λ x}

Where λ is the rate parameter and λ = 1/ (average time between events).

The PDF of a chi-square distribution is given by,

f (x) = \frac{x^{\frac{p}{2} - 1} e^{- \frac{x}{2}}}{2^{\frac{p}{2}} Γ (\frac{p}{2})}

Where p denotes the number of degrees of freedom and Γ denotes the gamma function.

The PDF of a t-distribution is given by,

f (t) = \frac{Γ (\frac{p + 1}{2})}{\sqrt{p π} Γ (\frac{p}{2})} {(1 + \frac{t^{2}}{p})}^{- (\frac{p + 1}{2})}

Where p denotes the number of degrees of freedom and Γ denotes the gamma function.

The chi-square value is determined by,

χ^{2} = \sum_{i} \frac{{(O_{i} - E_{i})}^{2}}{E_{i}}

Where O denotes the actual value and E denotes the predicted value.

The t-statistic is evaluated by,

t = \frac{\bar{x} - μ}{\frac{s}{\sqrt{n}}}

Where x̄ represents the sample mean, μ represents the population mean, and s represents the sample variance.

Context and Applications

In statistical quality control, this continuous probability distribution has a wide range of applications. This form is commonly used in the analysis of a large sample when normality is a factor. The curves of this type of continuous probability distribution are ideal for studying sample statistics. A continuous probability distribution can be transformed even if the variable is not following.

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