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Exam Probability and Statistics Bernardo Guimaraes September 2007 (9 pts) A researcher collected 100 observations on a random variable X and calculated the mean and the median He obtained M ean(X) = 10 and M edian(X) = Subsequently, he realized that the highest observation in his sample was misreported He had x1 = 20, but the correct value was x1 = 21 (a) Using the correct value of x1 , what is the mean of X? If it can’t be calculated, explain why (b) Using the correct value of x1 , what is the median of X? If it can’t be calculated, explain why (c) What the values of M ean(X) and M edian(X) hint about asymmetry? (7 pts) Let X and Y be random variables with joint probability density function fX,Y (x, y) = for ≤ x ≤ 1, ≤ y ≤ and ≤ x + y ≤ Are X and Y independent? Explain (11 pts) X and Y are independent random variables, with marginal density functions: x , 0≤x≤2 fY (y) = , ≤ y ≤ fX (x) = Find the joint cumulative distribution function for all possible values of x and y (9 pts) A supermarket has express lines Let X and Y denote the number of customers in the first and the second, respectively, at a given time Denote by pX,Y (x, y) the joint probability function of X and Y (a) Would you expect X and Y to be independent? (b) Would you expect pX,Y (x, y) to be symmetric, that is, would you expect pX,Y (a, b) = pX,Y (b, a)? (c) Would you expect pX,Y (5, 2) to be bigger than pX,Y (2, 2)? Or bigger than pX,Y (5, 5)? (7 pts) Let X and Y be random variables such that Y = 2X + The probability density function of X is: fX (x) = , ≤ x ≤ What is the probability density function of Y ? (10 pts) Consider the random variable Y , such that Pr(Y = 1) = θ, and Pr(Y = 0) = − θ A sample consisting of n independent realizations of Y ({y1 , y2 , , yn }) was collected Find the method of moments estimator of θ (11 pts) Consider the random variable X, such that Pr(X = x) = (1 − p)x−1 p, for x = 1, 2, A sample consisting of n independent realizations of X ({x1 , x2 , , xn }) was collected Find the maximum likelihood estimator of p (11 pts) The time between eruptions of a given volcano, T , follows an exponential distribution, with hazard rate λ (see formulas below) A researcher wants to test whether λ < 5, with level of significance 5% For that, he has only one observation, t (a) What are the null and alternative hypotheses? (b) What are the values of t that lead the researcher to reject the null hypothesis? (12 pts) X is a random variable described by an exponential distribution A researcher collected a sample consisting of 81 independent realizations of X ({x1 , x2 , , x81 }) She found that: 81 81 X X xi = 6480 and (xi − x ¯)2 = 6480 i=1 i=1 where x ¯ is the sample mean To simplify calculations, note that 6480 = 80 × 81 Find a confidence interval for the mean of the distribution (µ) with 95% confidence 10 (13 pts) Consider a normal random variable with unknown mean µ and variance σ = A researcher wants to test whether µ is greater than (a) What are the null and alternative hypotheses? (b) At 5% level of significance, what should be the size of the sample if the researcher wants to reject the null hypothesis with probability 90% if µ = 1? • You can apply the Central Limit Theorem for n > 30 √ √ √ • For calculations, you can use: = 1.4, = 1.7 and = 2.2 • Exponential distribution (hazard rate λ): — Probability density function: f (t) = λe−λt — Cumulative distribution function: f (t) = − e−λt — Expected value: µ = λ • Normal distribution: — Probability density function: f (x) = x−µ √ e− ( σ ) 2πσ 2