110年 - 110 國立中山大學碩士暨碩士專班招生考試_應數系碩士班/甲組:機率與統計#105484

1. [10%] The probability density function of a random variable Z is
Find the cumulative distribution function of S = Z².

4. [10%] An online poll states that 50% people dislike a certain policy. For the next poll on the same topic, let be the point estimate of the true proportion ±2.5%? (Note:is the cumulative distribution function of a standard normal random variable Z. And 1.96² = 3.8416, 1.64² =2.6896, 1.28² =1.6384.)

5. [16%] Fill in the cells ⓐ~ⓔ on the incomplete analysis of variance table below from a one way randomized design. Also write down ⓕ the null hypothesis, ⓔthe alternative hypothesis, and ⓗ the conclusion for the F test using a significance level of 5%.

7. [10%] Suppose that a vector a consists of two random variables as x = (X₁, X₂)T, whose 2 x 2 variance-covariance matrix is Let w = (w1, w2)T be a given vector of known weights. Prove that the variance of the random variable

a. [8%] Find sufficient statistics for θ.