96年 - 96 淡江大學轉學考機率與統計學#56036

科目：轉學考-機率與統計學 | 年份：96年 | 選擇題數：0 | 申論題數：11

試卷資訊

所屬科目：轉學考-機率與統計學

選擇題 (0)

申論題 (11)

1, (10 points) Show that a random variable is called "memoryless” if and only if it is an exponential random variable.

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?

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 a^th event occurs, find the distribution function of X and its moment generating function.

(a) Find the value of c

(b) Find the marginal p. d f. of X.

【已刪除】(c) Find

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.

(a) Find the maximum likelihood estimator of θ .

(b) Does that MLE of θ converges to θ in probability (it is a consistent estimator)? Prove it, if it does.

【已刪除】

【已刪除】7.