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96年 - 96 淡江大學 轉學考 機率與統計學#56036
> 申論題
1, (10 points) Show that a random variable is called "memoryless” if and only if it is an exponential random variable.
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2，(10 points) Suppose that airplane engines will fail, when in flight, with probabi lity (l - p) independently from engine to engine. If an airplane needs a majority of its engines operative to make a successful flight, for what values of p is a 4-engine plane preferable to a 2-engine one?
#212548
3. (20 points) Suppose that the number of events occurs in a small interval follows from a Poisson process with parameter ②.Let X denote the waiting time until the ath event occurs, find the distribution function of X and its moment generating function.
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(a) Find the value of c
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(b) Find the marginal p. d f. of X.
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5. (15 points) Let the random vector (X， Y) has a bivariate normal distribution. Show that the best linear predictor of Y with respect to X is given by E[Y|X]. [Note]: For any real-valued functions, g and h，we say that the predictor g(X) is a better estimator of Y than h(X) does，if the m. s. e. of g(X) is less than that of h(X), that is E[Y-g(X)]2 ≤E[Y -h(X)]2.
#212553
(a) Find the maximum likelihood estimator of θ .
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(b) Does that MLE of θ converges to θ in probability (it is a consistent estimator)? Prove it, if it does.
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(b) Find K and K(2).
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(a) Find power function K(θ), θ ＞ 0.
#424594
(b) Find dxdy.
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