110年 - 110 國立臺灣大學_碩士班招生考試_應用數學科學研究所數理統計組:機率統計#102196

1. (8%) (7%) Let 'T be a failure time with support {t₁， t₂,...， t_n，...，) and in terms of λj's, i=l,...，...，n，...

2. (15%) Let X and Y be mutually independent and continuous random variables with the corresponding probability density functions fx(x) and fY(y). Derive the probability density function of Y conditioning on X -Y = 0.

3. (7%) (8%) Let X = Rcosθ and Y a Rsin θ, where R is a positive random variable on (0,∞), θ is a uniform random variable on (0, 2π), and R and θ are mutually independent. Derive the corresponding distributions of X/Y and 2XY/

4. (15%) Let U₁,...,U_n,... be independent randon variables from Uniform(0, 1) and P(X = x) = (1/(e - 1)x!) I(1,2,3,...,)(x) be the probability density function of X. Find the probability density function of Z = minf{U₁,...,Ux}.

5. (7%) (8%) Let X₁,..., X_n, be a random sample from Uniform(θ, θ + 1). Find a minimal suffcient statistic of θ and derive its distribution.

6. (15%) Let X₁,..., X_n be a random sample from N(u，σ²), where μ andσ² are unknown parameters. Derive the power function of the size a likelihood ratio test for the hypotheses H₀ : μ ≤ μ₀ versus H_A : μ ＞ μ0.

7. (10%) Let X1...., Xn be a random sample from a geometric distribution P(X = x) = and p have a uniform prior distribution on (0,1). Find the Bayes estimator of p based on the loss function

110年 - 110 國立臺灣大學_碩士班招生考試_應用數學科學研究所數理統計組:機率統計#102196

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