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Continuous Distribution

Any variable becomes a continuous if it is able to take any values between two specified values, otherwise it is called discrete variable, where all the possible values will be at most countable. In statistics, any probability distribution can mainly be divided into mainly two of distribution. They are as follows:

  1. Discrete Probability Distribution
  2. Continuous Probability Distribution

The probability distribution whose random variable falls under the continuous variable set is called continuous probability distribution.

Some of the continuous probability distribution examples are
  • normal distribution
  • chi-squared distribution
  • Uniform distribution,etc.

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The probability density function for the normal distribution can be represented by the equation

f(x) = $\frac{1}{\sigma \sqrt{2 \pi}}$ $e^{\frac{1}{2} \left(\frac{x - \mu}{\sigma} \right)}$

Normal distribution is in the shape of a bell and depends on the parameter. The most important aspect of this function is that what ever be the values of the parameters, $\mu$ and $\sigma$,the whole curve is divided into different sections of area that will be true for all types of normal curves.

Standard Normal Distribution


Any normal distribution can be converted to standard form by making the following changes ,

Z = $\frac{x - \mu}{\sigma}$
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Continuous Uniform Distribution

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The continuous uniform distribution is called as rectangular distribution, as it attains a rectangular shape. If a continuous random variable X can assume any value in the interval a $\leq$ x $\leq$ b and only these values, and if its probability density function f(x) is constant over that interval and equal to zero, then X is said to be uniformely distributes and its distribution is called a continuous uniform probabiltiy distribution. The probability density function that defines this distribution called the uniform probability density function. Its probability density function is defined as

f(x) = $\left\{\begin{matrix}
\frac{1}{b - a} & a < x < b \\
0& otherwise
\end{matrix}\right.$

Mean and Variance of Continuous Uniform Distribution


Mean value of this distribution is $\frac{(a + b)}{2}$

The variance of this distribution is $\frac{(b - a)^{2}}{12}$

The graphical representation of the function will be as follow:


Continuous Probability Distribution

Here, all intervals of the same length on the distribution are equally probable. Here, there will be always two parameters, say a and b, where a can be taken as minimum and b as maximum. So, they form the boundary values.

Cauchy Distribution

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The general form of the probability density function of the Cauchy distribution is given as:

f(x) = $\frac{\mu}{\pi}$ $\left(\frac{1}{\mu^{2} + (x - \theta^{2})}\right)$

where $\theta$ is the location parameter and $\mu$ is the scale parameter.

When $\theta$ = 0 and $\mu$ = 1, this equation can be transformed into the standard Cauchy distribution, and hence we obtain

f(x) = $\frac{1}{\pi}$ $\left(\frac{1}{1^{2} + (x^{2})}\right)$

The probability density function of the Cauchy distribution can be drawn as follows:

Continuous Distribution
In Cauchy distribution, the mean and the variance is undefined. Both the median and the mode will be the local parameter $\theta$.

Cauchy Distribution Characteristics:

1. A random variable X will have the Cauchy distribution if and only if the mean of the sample has a Cauchy distribution.
2. The properties of linear combinations of Cauchy variables characterize the Cauchy distribution.
3. If for some real a, which is not a tangent of a rational multiple of $\pi$, $\frac{1 + aX}{x - X}$ is distributed the same as X, then X is Cauchy with probability density function
C (1, 0).
4. A necessary and sufficient condition for the random variable X to be Cauchy distributed with probability density function C(1, 0) is that for every -$\infty$ < y < z < $\infty$ there will be
$E\left[\frac{2X}{y < X \leq z} \right]$ = $\frac{\log(\frac{1 + x^{2}}{1 + y^{2}})}{\arctan(\frac{z-y}{1 + zy})}$

The Cauchy distribution is an example of a pathological case. It is very similar in shape to the normal distribution. The only difference is that their tails are much heavier when compared with the normal. When any test is studied using the normality, it would be good to check the test using the Cauchy distribution in order to see how sensitive it will be as it moves to wards the heavy tail part.

Fishers Z-Transformation

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Fishers Z-Transformation is defined as a procedure in which the product-moment correlation coefficient is modified into an interval scale which is not bounded by +1.00. Some of its main use is that it may be used to
  • Test the null hypothesis to check if a calculated correlation is significantly different from some hypothesized value.
  • Test the significance of the difference between two independent correlations
  • Find the average of several correlations
  • Form a confidence interval for a correlation coefficient, etc

Continuous Probability Distribution

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For a continuous probability distribution, the probability that it will assume a single or particular value will be zero. Hence it can never be written in a discrete tabular form. Mostly an equation will be used to express the continuous probability distribution. The equation that will be used here is called probability density function.
  • If f(x) is a continuous probability density function defined over a specified domain of x, then the area under the curve obtained by drawing the function = $\int_{-\infty}^{\infty}$ f(x) dx = 1
Also P(a < x < b) = $\int_{a}^{b}$ f(x) dx defines the area under the curve defined by x = a and x = b.
  • If X is a continuous random variable, then it has a probability density function, and so its probability will be between a given interval[a, b], such that
P(a < x < b) = $\int_{a}^{b}$ f(x) dx

Continuous Probability Distribution Definition


If a random variable is a continuous variable, its probability distribution is called a continuous probability distribution. The normal distribution is the most frequently used continuous probability distribution in statistics. A continuous random variable is the one which can take a continuous range of values, where the set of possible values for the random variable is at most countable. The equation used to describe a continuous probability distribution is called a probability density function (pdf). All probability density functions satisfy the following conditions:
  • The random variable Y is a function of X, y = f(x).
  • The value of y is greater than or equal to zero for all values of x, y $\geq$ 0 for all x.
  • The total area under the curve of the function is equal to one.

Continuous Probability Distribution Example

Given below is an example of continuous probability distribution.

Solved Example

Question: X is the continuous variable, the mass, in kg, of a substance produced per minute in an industrial process, where

f(x) = $\left\{\begin{matrix}
 \frac{6}{x^2} x(6 - x)& x \geq 5 \\
 1& otherwise
\end{matrix}\right.$ Find the probability that the mass is more than 5 kg.
Solution:
 
f(x) = $\frac{3}{x^{2}} for x $\geq 5$
The probability that x is greater than 5 will be the integral of probability function between infinity and 5.
Hence, probability P($X \geq 5$) = $\int_{5}^{\infty}\frac{3}{x^{2}} = \frac{-3}{x}_{5}^{\infty}= \frac{3}{5}$

 

Continuous Poisson Distribution

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The Poisson distribution gives the probabilities of various numbers of random events in a given interval of time when the possible number of discreate events is larger than the average number of events in the given interval. The Poisson distribution for continuous event numbers k. This can be achieved by replacing the factorial by the gamma-function:

$P_{\lambda}$ (k) = $\frac{\lambda^k e^{-\lambda}}{k!}

= \frac{\lambda^k e^{-\lambda}}{\Gamma (k + 1)}$.

Continuous Binomial Distribution

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The binomial distribution is used probability distribution of a discrete random variable. It plays a major role in quality control and quality assurance function. But the continuous version of the Binomial distributions is defined as, continuous binomial distribution with the parameters y > 0, 0 < p < 1 we will mean the probability measure with

F(x) = $\frac{B_p(x, y + 1 - x)}{B(x, y + 1 - x)}$ * 1(0 < x < y + 1) + 1(x ≥ y + 1)

where, x belongs to real numbers.
More topics in Continuous Distribution
Uniform Distribution Gamma Distribution
Beta Distribution Exponential Distribution
Weibull Distribution
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