Among 33 students in a class, 17 of them earned A's on the midterm exam, 14
earned A's on the final exam, and 11 did not earn A's on either examination.
What is the probability that a randomly selected student from this class earned an A
on both exams?
Suppose that three numbers are selected one by one, at random and without
replacement from the set of numbers {1, 2,3,...,n} . What is the probability that the
third number falls between the first two if the first number is smaller than the second?
Prove that if X is a positive, continuous, memoryless random variable with
distribution function F , then F(t) = 1− e−λt , for some λ > 0 . This shows that the
exponential is the only distribution on (0,∞) with the memoryless property.
Let X1, X2 , X3 , and X4 be four independently selected random numbers from
(0,1). Find P(1/ 4 < X(3) < 1/ 2) . X(3) is the 3rd smallest value in {X1, X2 , X3 , X4}.
Mark each of the following statements True (T) or False (F). (Need NOT to give reasons.)
(1) A real square matrix may have complex eigenvalues and complex eigenvectors.
(2) Let M be a symmetric matrix. If M is invertible, then M−1 is also a
symmetric matrix.
(3) Let M be a real square matrix of size n . If || Mx ||2 =|| x ||2 for all x∈n , then
M is an orthogonal matrix, MTM = In .
(4) Let M be an m×n matrix, m ≠ n . We have rank(MTM) = rank(MMT ) .
(5) Let M be an m×n matrix, m ≠ n . We have nullity(MTM) = nullity(MMT ) .
Let Im and In be identity matrices of sizes m and n , respectively, where we
assume m > n . Can you find an m× n matrix A and an n×m matrix B such
that AB = Im and BA = In ? (Explain your answer)
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