Subsequently, he realized that the highest observation in his sample was misreported.. a Using the correct value of x1, what is the mean of X?. If it can’t be calculated, explain why.. b
Trang 1Exam Probability and Statistics
Bernardo Guimaraes September 2007
1 (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) = 8 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 do the values of M ean(X) and M edian(X) hint about asymmetry?
2 (7 pts) Let X and Y be random variables with joint probability density function
fX,Y(x, y) = 2 for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and 0 ≤ x + y ≤ 1 Are X and Y independent? Explain
3 (11 pts) X and Y are independent random variables, with marginal density functions:
fX(x) = x
2 , 0 ≤ x ≤ 2
fY(y) = 1 , 0 ≤ y ≤ 1 Find the joint cumulative distribution function for all possible values of x and y
4 (9 pts) A supermarket has 2 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)?
5 (7 pts) Let X and Y be random variables such that Y = 2X + 2 The probability density function
of X is:
fX(x) = 1 , 0 ≤ x ≤ 1 What is the probability density function of Y ?
6 (10 pts) Consider the random variable Y , such that Pr(Y = 1) = θ, and Pr(Y = 0) = 1 − θ
A sample consisting of n independent realizations of Y ({y1, y2, , yn}) was collected Find the method of moments estimator of θ
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Trang 27 (11 pts) Consider the random variable X, such that Pr(X = x) = (1 − p)x −1p, 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
8 (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?
9 (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
X
i=1
xi= 6480 and
81
X
i=1
(xi− ¯x)2= 6480
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 σ2 = 4 A researcher wants to test whether µ is greater than 0
(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: √2 = 1.4,√
3 = 1.7 and√
5 = 2.2
• Exponential distribution (hazard rate λ):
— Probability density function: f (t) = λe−λt
— Cumulative distribution function: f (t) = 1 − e−λt
— Expected value: µ = 1
λ
• Normal distribution:
— Probability density function: f (x) = √1
2πσe−1(x−µσ )2.
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