Random Variables, PDFs and CDFs

The concept of probability is very important in the field of electrical engineering, where outcomes can be nondeterministic. In a nondeterministic outcome, an experiment can be repeated multiple times and have different outcomes. For example, in a communication link messages may not be delivered the same way each time. Another example would be the failure of manufactured parts even within a scheduled lifetime.

A random variable is defined as a function that maps a real number to an outcome within a sample space (a set that contains all possible outcomes of a random experiment). The “real number” is sometimes called an observation. The range includes all the possible observations and the domain includes all of the possible outcomes of the experiment. A single random variable produces a single observation. Random variables are notated by capital letters generally towards the end of the alphabet (eg. X(a), U(a), etc).

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A useful function for describing probability of random variables is the Cumulative Distribution function (CDF). This function is defined as the probability that the random variable is less than a certain value, x.

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Setting x = infinity shows that the CDF should equal to one. This is equivalent to the probability of the sure event (which is always one because it is the probability of the entire sample space). Setting x = 0 gives a probability of 0 (probability of the null event is always zero). The value of the CDF must always be between these two values and must never decrease. The CDF can be discontinous (in the case of discrete random variables) as well as continuous (in the case of continuous random variables).

Finding the probability between two values is easily obtained from the CDF.

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Taking the derivative of the CDF leads to the PDF (probability density function).

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This shows that the PDF and CDF are inverse functions. Alternatively, the PDF can be defined as

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This is because the area under the PDF is what determines probability. The total integration of the PDF must equal one.