Now, depending on the support set of the random variable $X$, the density (or mass function) need not exist. For any type of random of random variable, the CDF always exists (and is unique), defined as $$F_X(x) = P\.$$ PMFs are associated with discrete random variables, PDFs with continuous random variables.
PMF TO CDF PDF
You can go from pdf to cdf (via integration), and from pmf to cdf (via summation), and from cdf to pdf (via differentiation) and from cdf to pmf (via differencing), so when you have a pmf or a pdf, it contains the same information as the cdf. It's difficult to answer the question 'do they contain the same information' because it depends on what you mean. To get probabilities from pdfs you need to integrate over some interval - or take a difference of two cdf values. The pdf doesn't itself give probabilities, but relative probabilities continuous distributions don't have point probabilities. The pmf for a discrete random variable $X$, gives $P(X=x)$. The cdf for a random variable $X$ gives $P(X\leq x)$
(such as a mixed distribution - for example, consider the amount of rain in a day, or the amount of money paid in claims on a property insurance policy, either of which might be modelled by a zero-inflated continuous distribution). The cdf applies to any random variables, including ones that have neither a pdf nor pmf * formal approaches can encompass both and use a single term for them Hope this worked out example with a touch of integration helps to refresh your calculus skills.Where a distinction is made between probability function and density*, the pmf applies only to discrete random variables, while the pdf applies to continuous random variables. Thus the following two functions are possible PDFs. The exponential, Weibull and other distributions have PDFs defined, yet it is possible to have an arbitrary function meet the requirements of a PDF.
Another property is the PDF is defined across the entire sample space. The PDF also has the property that the area under the curve for is one. Like a histogram, the PDF when plotted reveals the shape of the distribution.
PMF TO CDF ISO
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