Mathematics Homework Help

1. Suppose that x1, x2, x3, x4, is a sample drawn from the uniform distribution f(x;θ):= 1/ θ if x∈[1,1+θ], or

0 otherwise,

where θ is an unknown parameter. Let y1, y2, y3, y4 be the associated order statistics.

(a) Find an expression for the joint likelihood function f(x1, x2, x3, x4; θ).

(b) Is y2 a sufficient statistic for θ? Give a brief explanation of your answer.

(c) Now suppose that x1 = 5, x2 = 3, x3 = 4, x4 = 2. Find the maximum likelihood estimate of θ.

(d) Using the values from part (c), find the method of moments estimate for θ. (Hint: there is only one parameter so you only need to compute the first moment).

2. Suppose that X1, . . . Xn are i.i.d. random variables with probability density function

f(x,θ) = θe^(−θx ) if x≥0 or 0 otherwise.

(a)Give an expression for the joint likelihood function f(x1, . . . , xn; θ).

(b) Show that the sample mean is a sufficient statistic for θ.

3. Suppose you collect a sequence of data points (x1,y1),…,(xn,yn) and you use least squares regression to find the values of α0 and α1 so that the line y = α1x + α0 gives the best match to the data. Show that if the data points already lie on a line y = mx+b i.e. (yi = mxi +b for i = 1, . . . , n), then least squares regression chooses the parameters α0 = b and α1 = m.

4. Suppose you are playing a random game that has three possible outcomes WIN, LOSE or DRAW. The probability of winning the game is an unknown parameter θ ∈ [0, 1], losing and drawing have equal probability 12 (1 − θ). Suppose you have a prior on the data that is given by h(θ) = 2θ.

(a) Suppose that it costs $5 to play the game, you get $15 dollars if you win, nothing if you draw, and you have to pay an additional $2 if you lose. Based on the prior distribution would you play the game? Explain.

(b) Now suppose you observe the following sequence of (independent) game out- comes: WIN, LOSE, DRAW, DRAW, WIN, LOSE, WIN, LOSE, DRAW, DRAW. Give an expression for the posterior distribution of θ.

(c) Based on the posterior distribution would you play the game?

(d) Would your answer to part (c) change if you had observed the following sequence of (independent) game outcomes instead: WIN, LOSE, LOSE, LOSE, WIN, LOSE, WIN, LOSE, LOSE, LOSE. Give a brief explanation of your reasoning.