TABLE OF CONTENTS The next examples of the use of conditional distributions treat the distributions arising when one random variable (thought of as a noise term) is added to another, independent random variable (thought of as a signal term). This is an important example of a derived distribution problem that yields an interesting conditional probability. The problem also suggests a valuable new tool which will provide a simpler way of solving many similar derived distributions the characteristic function of random variables.
Consider two independent random variables X and W and form a new random variable Y = X + W. This could be a description of how errors are actually caused in a noisy communication channel connecting a binary information source to a user. In order to apply the detection and classification signal processing methods, we must first compute the appropriate conditional probabilities of the output Y given the input X. To do this we begin by computing the joint pmf of X and Y using the inverse image formula:
Note that this formula only makes sense if y ? x is one of the values in the range space of W. Thus from the definition of conditional pmf s,
an answer that should be intuitive: given the input is x, the output will equal a certain value y if and only if the noise exactly makes up the difference, i.e., W =
