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2.3 Continuous Random Variables

So far we have considered only discrete r.v.'s, that is, a specific number is assigned to the event , but what if the events cannot be enumerated by integers, such as the angle of scattering for an electron scattering off a gold nucleus or the time to failure for a computer chip? The above definitions for discrete r.v.'s can be easily generalized to the continuous case.

First of all, if there is a continuous range of values, such as an angle between 0 and , then the probability of getting exactly a specific angle is zero, because there are an infinite number of angles to choose from, and it would be impossible to choose exactly the correct angle. For example, the probability of choosing the angle radians must be zero, since there are an infinite number of alternative angles. In fact, there are an infinite number of angles between 1.33 and 1.35 radians or between 1.335 and 1.345 radians, hence the probability of a given angle must be zero. However, we can talk about the probability of a r.v. taking on a value within a given interval, e.g., an angle between 1.33 and 1.35 radians. To do this, we define a probability density function, or pdf.