STT 441 Final Exam Fall 2013

Transcription

1 STT 441 Final Exam Fall 2013 (12:45-2:45pm, Thursday, Dec. 12, 2013) NAME: ID: 1. No textbooks or class notes are allowed in this exam. 2. Be sure to show all of your work to receive credit. Credits are given to your reasoning and solution, not to the final answer. Please write clearly, in order and to the point. 3. Relax and GOOD LUCK! 1

2 1. Assume that X is uniformly distributed over (0,3). (a) Find the cumulative distribution function of X. [7pts] (b) Find the density function of Y = e X. [7pts] 2

3 2. You ask your neighbor to water a sicky plant while you are on vocation. Without water, it will die with probability 0.9; With water, it will die with probability 0.1. You are 80 percent certain that your neighbor will remember to water the plant (a) What is the probability that the plant will be alive when you return? [7pts] (b) If the plant is dead upon your return, what is the conditional probability that your neighbor forgot to water it? [7pts] 3

4 3. If X 1, X 2 and X 3 are independent normally distributed random variables, each having mean 0 and variance 1. Let Y 1 = X 1 2X 2 and Y 2 = X 2 + 2X 3. (a) What is the marginal distributions of Y 1 and Y 2? Specify the distributions and the corresponding parameters. (b) Compute the covariance between Y 1 and Y 2. 4

5 4. The joint probability mass function (pmf) f(x, y) of (X, Y ) is given in the following table y x /4 1/ /8 1/8 1/4 (a) Fill the marignal pmfs f X (k) and f Y (k) of X and Y, respectively in the following tables k 3 4 f X (k) k f Y (k) (b) What is the distribution of X 3? Give the name and the corresponding parameters. (c) Are X and Y independent? Give your reason. 5

6 (d) What is the pmf of Z = X + Y? (e) Find the variance of 2X + Y. 6

7 5. The random variables X and Y have a joint density function given by f(x, y) = { 2 x exp( 2x) 0 < x <, 0 < y x 0 otherwise (a) Find the marginal density of X. (b) Find the conditional density of Y given X = 2. 7

8 (c) Find the conditional expectation of Y 2 + 2X given X = 2. (d) Find the expectation of Y. (e) Find the probability P (X + Y 8, X 4). 8

9 Possibly useful formula 1. P (A c ) = 1 P (A); 2. P (A B) = P (A) + P (B) P (A B) 3. If S is finite and each one point in the set is assumed to have equal probability, then P (A) = A / S, where E denotes the number of outcomes in the event E and S is the sample space. 4. If P (F ) > 0, then P (E F ) = P (EF )/P (F ). 5. P (E) = P (E F )P (F ) + P (E F c )P (F c ). 6. Bayes formula: P (E F ) = P (F E)P (E) P (F E)P (E) + P (F E c )P (E c ). 7. If events A and B are independent, then P (AB) = P (A)P (B). 8. For a random variable X, the CDF function of X is defined by F (x) = P (X x) for any x. 9. The probability mass function (pmf) of a discrete random variable X is f(x) = P (X = x) 10. The expectation of a discrete random varible X with pmf f(x) is E(X) = x xf(x) and the variance is Var(X) = E{(X E(X)) 2 } = E(X 2 ) E 2 (X). 11. A discrete random variable X is called a Binomial(n, p) random variable if the pmf f(x) is given by f(x) = ( n x) p x (1 p) n x for x = 0,, n. In particular, if n = 1, the random variable X is called Bernoulli(p). Its mean and variance is given by E(X) = np and Var(X) = np(1 p). 12. A discrete random variable X is called a Poisson(λ) random variable if the pmf f(x) is given by f(x) = λ x e λ /x! for x 0. Its mean and variance is given by E(X) = Var(X) = λ. 13. A discrete random variable X is called a Geometric(p) random variable if the pmf f(x) is given by f(x) = p(1 p) i 1 for i = 1, 2, 3,. Its mean and variance is given by E(X) = 1/p and Var(X) = (1 p)/p A random variable X is continous if there is a non-negative function f(x), called probability density function of X, such that for any set B, P (X B) = B f(x)dx. 15. If X is continous, then d dxf (x) = f(x). 16. If X is continous with pdf f(x), then the expectation of X is E(X) = xf(x)dx and Var(X) = E{(X E(X)) 2 } = E(X 2 ) E 2 (X). 17. A random variable X is said to be Uniform over the interval (a, b) if the pdf f(x) is given by f(x) = 1/(b a) for a x b and 0 otherwise. Its mean and variance is given by E(X) = (a + b)/2 and Var(X) = (b a) 2 /12. 9

10 18. A random variable X is said to be normal with parameters µ and σ 2 if the pdf f(x) is given by f(x) = (1/ 2πσ) exp{ (x µ) 2 /(2σ 2 )} for < x <. Its mean and variance is given by E(X) = µ and Var(X) = σ 2. If X is normal with mean µ and variance σ 2, then Z = (X µ)/σ is a standard normal with mean 0 and variance A random variable X is said to be exponential(λ) if the pdf f(x) is given by f(x) = (1/λ) exp{ x/λ} for 0 < x <. Its mean and variance is given by E(X) = λ and Var(X) = λ A gamma function Γ(α) = 0 x α 1 e x dx. It is known that Γ(α) = (α 1)Γ(α 1) for α > The joint cumulative probability distribution function of the pair of random variables X and Y is defined by F (x, y) = P (X x, Y y). 22. If X and Y are both discrete random variables, then their joint probability mass function is defined by f(x, y) = P (X = x, Y = y). 23. If X and Y are both discrete random variables having joint pmf f(x, y), then the marginal probability mass function of X can be obtained by f X (x) = y f(x, y). 24. If X and Y are both continuous random variables having joint pdf f(x, y), then P ((X, Y ) C) = C f(x, y)dxdy. 25. If X and Y are both continuous random variables having joint pdf f(x, y), then the marginal probability density function of X can be obtained by f X (x) = f(x, y)dy. 26. If X and Y are random variables having joint pdf/pmf f(x, y), then the conditional pdf/pmf of X given Y = y is f(x Y = y) = f(x, y)/f Y (y), where f Y (y) is the marginal pdf or pmf of Y. 27. If X and Y are having joint pdf/pmf f(x, y), then E[g(X, Y )] = x y g(x, y)f(x, y) if (X, Y ) are discrete and E[g(X, Y )] = g(x, y)f(x, y)dxdy. 28. For any random variables X 1,, X n, E(X X n ) = E(X 1 ) + + E(X n ). 29. The covariance between X and Y is Cov(X, Y ) = E(XY ) E(X)E(Y ). Cov( n i=1 X i, m j=1 Y j) = m j=1 Cov(X i, Y j ). n i=1 30. Var( n i=1 X i) = n i=1 Var(X i) + i j Cov(X i, X j ). 31. If the conditional pdf/pmf of X given Y = y is f(x y), then the conditional expectation of X given Y = y is E(X Y = y) = x xf(x y) when (X, Y ) are discrete and E(X Y = y) = xf(x y)dx when (X, Y ) are continuous. 32. E(X) = E[E(X Y )] and Var(X) = E{Var(X Y )} + Var{E(X Y )}. 10