13. Let the density function of X be

 Find the density functions of and Z=e^-2X. (10 points)

11 . A stack of cards consists of seven red and five blue cards. A second stack of cards consists of nine red cards. A stack is selected at random and three of its cards are drawn.If all of them are red, what is the probability that the first stack was selected? (Show your calculation steps.) (10 points)

12.(a) If two fair dice are rolled 4 times, what is the probability of at least one 6 (on either die) in exactly two of these 4 rolls? (5 points) (b) Suppose that, on average, in every two pages of a book there is one typographical error.What is the probability of at least one error on a specific page of the book? (5 points)

(a) Determine the value of c. (4 points)

(b) Find the marginal probability density functions of Y. (6 points)

(a) If X is a normal random variable with parameters μ = 3 and σ2 =16,find P(1＜X＜11).(4 points)

(b) If 48 random numbers are selected independently from the interval (0, 1),what is the approximate probability that the sum of these numbers is at least 20? (6 points)

101年 - 101 國立交通大學_碩士班考試入學試題_資訊聯招：線性代數與機率#113290