Why Cant We Directly Find The PDF Of The Transformation Of Random?

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Why can't we directly find the PDF of the transformation of random variables, say g(X) from the random variable X. Why do we have to first convert PDF into CDF and then have to differentiate to get to the PDF of the transformed random variable?

Cumulative distribution functions describe real random variables. Suppose that X is a random variable that takes as its values real numbers. Then the cumulative distribution function F for X is the function whose value at a real number x is the probability that X takes on a value less than or equal to x. \qquad F(x)=P(X{\leq}x) The standard abbreviation for “cumulative distribution function” is c.d.f. Each c.d.f. F has the following four properties. F is a nondecreasing function. F is right continuous. \displaystyle\lim_{x\to\infty}F(x)=1. \displaystyle\lim_{x\to-\infty}F(x)=0. Conversely, any function F with those four properties is the c.d.f. of a real random variable. The most useful random variables are either discrete or continuous, but there are random variables that are neither discrete nor continuous. For a discrete random variable, the c.d.f. is a step function. Here are some graphs of some c.d.f.’s of some binomial distributions Source. Wikimedia commons.Binomial distribution cdf.png For a continuous random variable the c.d.f. is differentiable. Here are some graphs of some c.d.f.’s of some normal distributions Source. Wikimedia commons. Normal distribution cdf.png

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Convert PDF: All You Need to Know

When a random variable is random within a certain range then we can say that it is normal if it is as smooth on a particular interval as if the interval were continuous. There are four main categories of normal distribution. In this talk we will introduce the n-dimensional, normal distribution. Then we will give several examples of the normal distribution and the application of the normal distribution to data to evaluate the probability of events. Then we will define the conditional probability and test it for two random variables x and Y. The conditional probability p(x|Y)=p(x|x). The sample standard deviation t is used as an approximation of the probability that the outcome at a particular time is equal to Y. The sample statistic is the probability that the outcome at that time is equal to y, i.e., p(x|Y) / p(y|Y).

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