Final answer:
The value of k cannot be determined without knowing the domain of x for the PDF f(x)=kx. The probability P(1 < x < 2) depends on the area under the PDF between x = 1 and x = 2, which would require integration to calculate if the domain were known.
Step-by-step explanation:
The question asks us to find the value of k when f(x)=kx is given as a probability density function (PDF) and to calculate the probability P(1 < x < 2) for this PDF. For a function to be a PDF, the total area under the curve from −∞ to +∞ must equal 1. However, kx is mentioned without limits, which makes it impossible to determine the value of k without additional information about the domain of x. Assuming there's a typo or missing information and that x has limits (0 to some number 'a'), we could find 'k' by setting the integral of f(x) from 0 to 'a' equal to 1 and solving for 'k'. Nonetheless, we cannot proceed without clarification on the domain of x. As for P(1 < x < 2), for any continuous PDF, P(x = c) = 0 because the probability over a single point, which has no width, is zero. Therefore, P(1 < x < 2) is the area under the graph of the PDF between x = 1 and x = 2, which we can find geometrically or using integration if the actual PDF and its domain are known.