Final answer:
The expected value of a continuous random variable X with probability density function (pdf) fX(x) is defined as E(X) = ∫ from −∞ to ∞ x ⋅ fX(x) dx, representing the weighted average of all possible values where the weights are the pdf values.
Step-by-step explanation:
The expected value of a continuous random variable, often symbolized as E(X) or the mean μ, is the long-term average value of repetitions of the same experiment. For a continuous random variable X with a probability density function (pdf) fX(x), the expected value is defined using integration over its entire range from negative to positive infinity. Mathematically, this is expressed as E(X) = ∫∞−∞ x · fX(x) dx. This integral sums up the product of each possible value of the random variable and its corresponding probability density, reflecting the continuum of values that X can take on.
To understand the concept, consider that the probability density function (pdf) describes the density of probability at each point within a range, and the area under the pdf curve over an interval represents the probability that X falls within that interval. Like the discrete case where we sum the products of values and their probabilities, in the continuous case, we instead integrate these products over all possible values of X to obtain the expected arithmetic average when an experiment is repeated an infinite number of times.
An analogy in a discrete setting is: if X represents the number of heads from three coin tosses, by using the discrete formula Σ xP(x), the expected value would reflect the average number of heads obtained in a large number of trials. Translating this to the continuous setting where the variable can take on an infinite range of values, integration is necessary to compute the analogous average or mean.