2024 Paper - I General Studies and Engineering Aptitude

The standard deviation of the exponential distribution of fₓ(x) = \begin{cases} \lambda e^{-\lambda x}, & x ≥ 0 \\ 0, & x < 0 \end{cases} is

Suppose that 0.1% of the people in a certain area have a disease D and that a mass screening test is used to detect cases. The test gives either a positive or a negative result for each person. Ideally, the test would always give a positive result for a person who has D, and would never do so for a person who has not. In practice the test gives a positive result with probability 99.9% for a person who has D, and with probability 0.2% for a person who has not. What is the probability that a person for whom the test is positive actually has the disease?

Let the random variables X and Y have joint density function given by f_{X,Y}(x,y) = \begin{cases} c(1-y), & 0 ≤ x ≤ y ≤ 1 \\ 0, & otherwise \end{cases} Then the marginal density function for X is

The continuous-time signal f(t) = e^{-2 \omega t}, where \omega is a real constant, is sampled when t ≥ 0 at intervals T. What is the z transform of the resulting sequence of samples?

If (z) = z/z² - z + 1, then the inverse z transform of Y(z) is

The temperature distribution T(x) at a distance x, measured from one end, along a bar of length L is given by T(x) = Kx(L-x)(0 ≤ x ≤ L), K = constant. A Fourier series expansion consisting of sine terms only for T(x) is