Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers, 3rd Edition

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Probability Models of Derived Random Variables

There are many situations in which we observe one or more random variables and use their values to compute a new random variable. For example, when voltage across an r₀ ohm resistor is a random variable X, the power dissipated in that resistor is Y = X²/r₀. Circuit designers need a probability model for Y to evaluate the power consumption of the circuit. Similarly, if the amplitude (current or voltage) of a radio signal is X, the received signal power is proportional to Y = X². A probability model for Y is essential in evaluating the performance of a radio receiver. The output of a limiter or rectifier is another random variable that a circuit designer may need to analyze.

Radio systems also provide practical examples of functions of two random variables. For example, we can describe the amplitude of the signal transmitted by a radio station as a random variable, X. We can describe the attenuation of the signal as it travels to the antenna of a moving car as another random variable, Y. In this case the amplitude of the signal at the radio receiver in the car is the random variable W = X/Y. Other practical examples appear in cellular telephone base stations with two antennas. The amplitudes of the signals arriving at the two antennas are modeled as random variables X and Y. The radio receiver connected to the two antennas can use the received signals in a variety of ways.

  It can choose the signal with the larger amplitude and ignore ...