5. Determine whether the matrix is invertible, and if it is, please compute its inverse. (15%)

二、線性代數部分： Let \( A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \). Find \( A^2, A^3, A^n \). (8%)

8. Let the probabilities for characters A, B, C, D, E, F to appear in acharacter string be \( \frac{5}{54} \), \( \frac{6}{54} \), \( \frac{7}{54} \), \( \frac{9}{54} \), \( \frac{10}{54} \), and \( \frac{17}{54} \)respectively. Findbinary codes for these characters such that when a characterstring of these characters is represented by your code the binarystring will be of minimal length. Is 010101010101010 a binarystring corresponding to a character string in your code? Describehow do you find the character string of a binary string in yourcode? [10%]

(1)Find E(Y), E(), and variance . (10%)

(2)Find the moment generating function (m.g.f.) of Y. (5%)

2. Let X and Y be two random variables with joint density function given byFind the conditional expectation, . (10%)

(1) Calculate the moment generating function (m.g.f.) of X.　(7%)

(2) Find the expectation and using m.g.f. of X.　(8%)

4. Let and be independent r.v.’s distributed as Poisson with mean and set .Show that the conditional p.d.f. of , given , is .　(10%)

二、統計部分： 1. Of the first 100 patients to undergo a new operation, 6 die. Can you calculate the95% CI for the probability of dying with this procedure? If so, calculate theinterval. If not, what information do you need? What assumptions must you make?(10%)

2. A new diet and exercise program has been advertised as remarkable way toreduce blood glucose levels in diabetic patients. Ten randomly selected diabeticpatients are put on the program, and the results after 1 month are given by thefollowing table:Do the data provide sufficient evidence to support the claim that the newprogram reduces blood glucose level in diabetic patients? Use α=0.05, and　\( t_{0.05,9} = 1.833 \)。. (15%)

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