Our first set of questions is related to some basic exercises from the required prerequisite background...

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Our first set of questions is related to some basic exercises from the required prerequisite background in mathematics (Calculus I and Discrete Math). You should feel very comfortable doing these. Each question has equal mark of [0.1 pts]. • Let f(2)=rexp(zª), what is the derivative of f(x) w.r.t z? Consider an unfair coin that has probability 3/4 of coming up heads. If the coin is flipped three times, what is the probability of seeing tails each time: P(tails, tails, tails) =? • I agree to let you flip the unfair coin (the same unfair coin as the previous problem) one time and pay you $1 if it comes up heads and $2 if it comes up tails. Let X be the number of dollars I pay you, which is a random variable. What is the expected value of X: E[X] =? 1 • Write the result of the following matrix-matrix multiplication. Your answer should be written in terms of u, v, a, and b. Suppose (Br+y) A=0, where A and B are both invertible nxn matrices and r and y are vectors in R. and 0 is a vector of all zeros. Use the properties of matrix multiplication, transpose, and inverse to show that r=-B'y. Show your work. Our first set of questions is related to some basic exercises from the required prerequisite background in mathematics (Calculus I and Discrete Math). You should feel very comfortable doing these. Each question has equal mark of [0.1 pts]. • Let f(2)=rexp(zª), what is the derivative of f(x) w.r.t z? Consider an unfair coin that has probability 3/4 of coming up heads. If the coin is flipped three times, what is the probability of seeing tails each time: P(tails, tails, tails) =? • I agree to let you flip the unfair coin (the same unfair coin as the previous problem) one time and pay you $1 if it comes up heads and $2 if it comes up tails. Let X be the number of dollars I pay you, which is a random variable. What is the expected value of X: E[X] =? 1 • Write the result of the following matrix-matrix multiplication. Your answer should be written in terms of u, v, a, and b. Suppose (Br+y) A=0, where A and B are both invertible nxn matrices and r and y are vectors in R. and 0 is a vector of all zeros. Use the properties of matrix multiplication, transpose, and inverse to show that r=-B'y. Show your work.