Final answer:
The student is asking about calculating the expectation E[X²] of a random variable X with an exponential distribution. The random variable X is non-negative, and its PDF indicates it is exponentially distributed with a rate parameter of 2. The expected value E[X²] for an exponential distribution can be computed using the formula 2/λ², yielding 0.5 in this case.
Step-by-step explanation:
The student is asking about finding the expectation E[X²] for a random variable X with the probability density function (PDF) given by fX(x)=2e−2xu(X), where u(X) is the unit step function. This type of problem typically involves using the characteristic function of the random variable or integrating the PDF multiplied by x².
A random variable X can be defined as a variable whose outcomes are determined by chance, with each outcome having a certain probability. Since the provided PDF includes the unit step function, it assures that X takes on non-negative values. The function 2e−2x defines the exponential distribution, indicating that X follows an exponential distribution with a rate parameter of 2 (X~ Exp(0.5)).
The expected value of X², E[X²], can be found by integrating the PDF multiplied by x² over all possible values of X. For an exponential distribution X~ Exp(λ), the formula for expectation is E[X²] = 2/λ², which in this case would result in E[X²] = 2/2² = 0.5.