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| 科目：工程數學 |
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1.
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(15%) Let X be a random variable of probability density function f(x), which has the following property : | |
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Consider a discrete random variable Z which is independent of X and has the probability mass function: Pr{Z=+1} = Pr{Z=-1} = 1/2. Suppose Y= ZX. | |
| (a) | Are Y and X uncorrelated? Why? | |
| (b) | Are Y and X independent? Why? | |
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(c) | Assume X is of normal distribution with mean 0 and variance 1. Show that Y is also normal with mean 0 and variance 1.Do (a) and (b) contradict the fact that uncorrelated jointly normal random variables are independent? |
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2.
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(15%) Consider the following matrix: | |
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| (a) | Find  . |
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| (b) | For any nonzero vector 
, define a(x) =  ,
where T denotes taking transpose. Find the maximum and minimum of a(x)
for all nonzero vectors x in  . |
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3.
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(15%) Consider two independent random variables X and Y. X is uniformly distributed on intervals [0, 10] and [30, 40]. Y is uniformly distributed on interval [10, 30]. Find the probability density function of X+Y. | |
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4.
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(5%) Consider an n×n real-valued matrix A. Which of the following statements are equivalent to "A is nonsingular?" (Proofs are not needed but no partial credit is available for this problem.) | |
| (a) | Ax = 0 has a solution 0 for x. | |
| (b) | A +2A+I is nonsingular, where I is the n×n identity matrix. | |
| (c) | A is a matrix representing some linear transformation. | |
| (d) | The column vectors of A span  . |
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(e) | A is a transition matrix with respect to some ordered basis to another ordered basis. |
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5.
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(15%) Consider a discrete random variable X taking values from {0, 1, 2,3, …}, i.e., X is of nonnegative integer value. Please | |
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show that X has the following memoryless property: | |
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if and only if X is of geometric distribution, i.e., | |
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6.
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(15%) Consider the vector space C[a, b] consisting of all continuous function on interval [a, b]. Let W be the subspace of C[a, b] spanned by 1, sinhx, coshx, and denote by D the differentiation operation on W. | |
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(a) | Find the transition matrix T representing the change of coordinates
from the ordered basis [1, sinhx, coshx] to the ordered basis  |
| (b) | Find the matrix A representing D with respect to the ordered basis [1, sinhx, coshx]. | |
| (c) | Find the matrix B representing D with respect to the ordered basis .
What is the relation between A, B, and T? Verify your answer. |
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7.
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(5%) Assume the time between the occurrence of two earthquakes is exponentially distributed with mean 30 years. Suppose the last earthquake occurred 10 years ago. What is the expectation of time waiting for the next earthquake? Justify your answer. | |
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8.
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(15%) Find the equation of the circle that gives the best least squares circle fit to the points (1, 0), (2, 2), (3, 0), and (2, -2). Justify your answer. | |
,
for m
{0, 1,
2, 3, …} and 0 < p < 1.--- END---