Sta347 Probability I
Selected Practice Problems for Final
Dec. 10, 2013
(1) Let X have the density function f (x) = cxe2x , 0 x < . And f (x) = 0 otherwise.
(a). Find the value of c.
(b). Give the mean and the variance of X.
(c). Give the moment generatin
Probability tries to explain structured randomness.
What is "structured"? It's a property of some event happening
converges as we repeat the experiment many many times.
Empirical proportion of heads
Number of tosses
This is limit inf
Think about a discount (interest) problem. Suppose that at each round, the interest
rate is xed at . We always calculate according to the value of stake at the rst
round.
In general, one stake at ith game is worth
stakes for the rst round.
Suppose he wins
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But in this class we have to assume lots of stuffs
Are independent.
Bernoulli distributions
Counting
Combinations. If we have n objects, we select p out of them and we do not care
about the order of them, then the total number of ways to select the p obje
Sta347 Probability I
Selected Practice Problems for Midterm
Oct. 12, 2013
(1) Problem 2 on Page 24 of the Textbook.
(2) Problem 4 on Page 36 of the Textbook.
(3) Show that if A and B are events, then
P (AB) = P (A) P (B) + P (AB).
(4) (a) Show that if A a