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Main points of this exam paper are: Mutually Independent, Sentence, Partition, Equal, Uniformly Distributed, Calculate, Random Variables
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Department of EECS - University of California at Berkeley EECS 126 - Probability and Random Processes - Fall 2008 Midterm 1: 10/09/
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1.1. Define “Random Variable”
1.2. Complete the sentence: A, B, C are mutually independent if and only if
1.3. Bayes’s Rule. Assume that {A 1 ,... , An} form a partition of Ω and that pm = P (Am) and qm = P [B|Am] for m = 1,... , n. Derive P [Am|B] in terms of p’s and q’s.
1.4. Assume that X is equal to 2 with probability 0.4 and is uniformly distributed in [0, 1] otherwise. Calculate E(X) and var(X). (Hint: Recall that var(X) = E(X^2 ) − E(X)^2 .)
1.5. Two random variables X, Y are related so that aX + Y = b for some real constants a and b. Given E(X) = μ, var(X) = σ^2 , express E(Y ) and var(Y ) in terms of μ and σ.
There is a series of mutually independent Bernoulli experiments that individually have probability p of success and probability (1 − p) of failure. These experiments are conducted until the rth success. Let X be the number of failures that occur until this rth^ success. The pmf of X is:
pX (k) =
k + r − 1 k
pr(1 − p)k, k ≥ 0
a) Justify the pmf.
b) Express E(X) in terms of p and r.
Show that if three events A, B, and C are mutually independent, then A and B ∪ C are indepen- dent.
Find the allowable range of values for constants a and b such that the following function is a valid CDF F (x) = 1 − ae−x/b^ if x ≥ 0 , and 0 otherwise.
For those values of a and b, compute P (− 2 < X < 10) where X is the associated random variable.
