Statistical Analysis in Physics — DU B.Sc. (Hons) Physics Previous Year Question Papers

Sr. Number of the Question paper : Attempt FOUR questions in all. Use of statistical tables are allowed. (a) (b) (c) (d) (e) (f) What do you understand by the term Probability Density Function (PDF)? State at least three properties of a PDF. How can the probability of an interval be obtained from a given PDF? A medical survey shows that 8% of a population suffers from a certain disease. A diagnostic test gives a positive result with probability 0.92 when the disease is present and gives a false positive result with probability 0.04 when the disease is absent. Find the probability that a person actually has the disease given that the test result is positive. State the Central Limit Theorem. Discuss the importance of this theorem important in statistics and machine learning. Two fair coins are tossed simultaneously. Let (X) denote the number of heads obtained on the first coin toss and (Y) denote the total number of heads obtained in both tosses. Find the covariance between (X) and (Y). If a joint probability density function is denoted by 𝑓𝑓 𝑋𝑋𝑋𝑋(𝑥𝑥, 𝑦𝑦), give the formula to derive marginal distribution function over the variable 𝑋𝑋 and 𝑌𝑌, when the range for continuously varying parameters 𝑥𝑥, 𝑦𝑦 are given by (−∞ ≤ 𝑥𝑥, 𝑦𝑦 ≤ ∞) What is meant by prior distribution in Bayesian statistics? If the likelihood follows a Poisson distribution, identify the conjugate prior and write its expression. (a) A discrete random variable X follows a Binomial distribution with parameters n and p. If its probability mass function is given by Using this representation, derive E (X) and Var (X) (b) What is meant by marginal probability distribution? Let X and Y be jointly continuous random variables having joint density function fxy (x, Y) = Ax2 + y, for0≤x ≤1,0 ≤y≤1 Find the value of A and P(0 ≤ X ≤! (a) A distribution consists of three components with frequencies of 200, 250 and 300, having means of 25, 10 and 15, and standard deviations of 3, 4 and 5 respectively. Determine the mean and standard deviation of the combined distribution. (b) Let two random events occurring together be recorded as X = 1, 5, 3, 8, 9, 6, 5, 3 and Y = 14, 11, 17, 19, 15, 11, 19, 14. Determine the covariance matrix, correlation matrix in this case and carry out the eigen-decomposition. (a) State and derive Bayes' theorem for continuous random variables. Explain the role of prior and likelihood in Bayesian inference. (b) In 1989, there were three candidates for the position of principal — Mr. Chatterji, Mr. Iyengar, and Dr. Singh - whose chances of getting the appointment were in the ratio 4:2:3, respectively. The probability that Mr. Chatterji, if selected, would introduce co-education in the college is 0.3. The corresponding probabilities for Mr. Iyengar and Dr. Singh are 0.5 and 0.8, respectively. If the college introduced co-education in 1990, find the probabilities that it was introduced by: