Final answer:
To find the probability distribution Pw(w) for the transformed variable w = -x, one needs the original distribution of x, and then applies the transformation to adjust the probability and density functions accordingly, utilizing the inverse relationship between the exponential function and the natural logarithm.
Step-by-step explanation:
The student's question pertains to determining the probability distribution of a transformed random variable. If x is a given random variable and w is defined as w = g(x) = -x, then to find Pw(w), we need to understand the relationship between x and w. When we consider the transformation g(x), we are essentially reflecting the probability distribution of x about the y-axis.
For instance, if we have a probability P(X < x) such as 0.70 = P(X < x), and the cumulative distribution function (CDF) 1 - e-0.5x, taking the inverse of this CDF gives x as a function of the probability.
In other words, when we use the natural logarithm, it allows us to solve for x explicitly. As an example, transforming the equation yields -0.5x = ln(0.30), or x = 2.41 minutes.
Understanding that the exponential and natural logarithm functions are inverses is crucial in these computations, as they allow us to express and solve equations in different forms (For example, ln(ex) = x and eln x = x). Additionally, knowing how to manipulate these functions to express inverse operations such as division can be demonstrated by expressions like x-n = 1/xn.
In conclusion, to find the distribution Pw(w), one would need the original distribution function of x, apply the transformation w = -x, and make the necessary adjustments to the probability and density functions.