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In data analysis transformation is the replacement of a variable by a function of that variable. for example, replacing a variable x by the square root of x or the logarithm of x. In a stronger sense, a transformation is a replacement that changes the shape of a distribution or relationship.
Suppose we are given a random variable X with density fX(x). We apply a function g to produce a random variable Y = g(X). We can think of X as the input to a black box, and Y the output. We wish to find the density or distribution function of Y .
Perhaps the most common probability distribution is the normal distribution, or "bell curve," although several distributions exist that are commonly used. Typically, the data generating process of some phenomenon will dictate its probability distribution. This process is called the probability density function.
Relationship between PDF and CDF for a Continuous Random VariableBy definition, the cdf is found by integrating the pdf. F(x)=x 22b 212 21ef(t)dt.By the Fundamental Theorem of Calculus, the pdf can be found by differentiating the cdf. f(x)=ddx[F(x)]Mar 9, 2021
The probability density function (pdf) and cumulative distribution function (cdf) are two of the most important statistical functions in reliability and are very closely related. When these functions are known, almost any other reliability measure of interest can be derived or obtained.
By definition, the cdf is found by integrating the pdf. F(x)=x 22b 212 21ef(t)dt. By the Fundamental Theorem of Calculus, the pdf can be found by differentiating the cdf. f(x)=ddx[F(x)]
A continuous random variable Z is said to be a standard normal (standard Gaussian) random variable, shown as Z 23cN(0,1), if its PDF is given by fZ(z)=1 21a2\u03c0exp{ 212z22},for all z 208R.
The concept is very similar to mass density in physics. its unit is probability per unit length. To get a feeling for PDF, consider a continuous random variable X and define the function fX(x) as follows (wherever the limit exists). fX(x)=lim\u0394 1920+P(x
( PD in PDF stands for Probability Density, not Probability.) f(\ud835\udc99) is just a height of the PDF graph at X = \ud835\udc99 However, a PDF is not the same thing as a PMF, and it shouldn't be interpreted in the same way as a PMF, because discrete random variables and continuous random variables are not defined the same way.
The transformation theorem provides a straightforward means of computing the expected value of a function of a random variable, without requiring knowledge of the probability distribution of the function whose expected value we need to compute.

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