所屬科目：研究所、轉學考(插大)◆統計學
(a) f(x) = for x = 0, 1, 2, 3, 4, 5
(b) f(x) = for x = 0, 1, 2, 3
(c) f(x) = for x = 3, 4, 5, 6
(d) f(x) = for x = 1, 2, 3, 4, 5
(a) Find the value of c that makes f(x, y) a probability density function.
(b) Find the marginal densities f₁(x) and f₂(y)
(c) Find the conditional densities f₁(x|y) and f₂(y|x)
3. (12分) Find the mean and the variance of the uniform probability distribution given byf(x) = for x = 1, 2, 3,..., n
(a) Find the maximum likelihood estimator of λ.
(b) Is the maximum likelihood estimator unbiased?
5. (12分) In a random sample of 400 industrial accidents, it was found that 231 were due at least partially to unsafe working conditions. Please construct a 99% confidence interval for the corresponding true proportion. (Refer to the attached Tables if necessary)
6. (10分) Let the variables X₁, X₂ and X₃ have variances σ₁² = 1.0, σ₂² = 3.0 and σ₃² = 5.0. Let these variables be independent and Y = 13 - 2X₁ + 3X₂ - 10X₃. Find the variance of Y.
(a) How much the degree of freedom will be when applying a sample of size n to estimate the model?
(b) What are 「R²」and 「adjusted R²」? And what is the difference between them?
(c) What is the multicollinearity problem?
(d) What is the heteroscedasticity problem?
(a) Show that if x ＜ α/(1 - β), then x ＜ E(Y) ＜
(b) Show that if x ＞ α/(1 - β), then x ＞ E(Y) ＞, and conclude that E[Y] is always between x and α/(1 - β).
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for x = 0, 1, 2, 3, 4, 5
for x = 0, 1, 2, 3
for x = 3, 4, 5, 6
for x = 1, 2, 3, 4, 5
for x = 1, 2, 3,..., n