MATH 254: Probability and Statistics II Spring 2023
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Mock midterm exam via the programme M¨obius for MATH 254: Probability and Statistics II Spring 2023
March 12, 2023
Question 1
We toss a fair repeatedly and independently. We let Xi = 1 if the outcome is heads at the i-th toss or 0 if tails. Define Sn = X1 + · · · + Xn .
a Determine E(Sn(2)) 1 point(s)
b Determine E(2Sn) 2 point(s)
c Determine the probability of the event {Sn = 1}. 2 point(s)
d Determine the probability of the event ╱n(l)
= ∞} 2 point(s)
e Determine the probability of the event n(l)
Xn exists} 2 point(s)
Question 2
a Which of the following statements are true?
(i) If X, Y are discrete random variables then so is XY. 1 point(s)
(ii) If X is a continuous random variable and Y is discrete with values 1, 2, . . ., then XY is continuous. 1 point(s)
(iii) The expectation of a random variable always exists. 1 point(s)
(iv) If X and Y are independent random variables then so are X + Y and X − Y. 1 point(s)
(v) If X is an absolutely continuous random variable with density f (x) such that f (x) > 0 for all x e R, then Y = min(X, 1) is continuous. 1 point(s)
b Which of the following statements is correct? 2 point(s)
(i) A Poisson(2) random variable has expectation 1/2.
(ii) If X and Y are Poisson(1) random variables then X + Y is Poisson(2).
(iii) The variance of a Poisson(λ) random variable is always 1/λ .
(iv) The variance of a Poisson(λ) random variable is λ .
(v) If X is Poisson(λ) then X/λ is always Poisson(1).
(vi) If X is Poisson(1) then X has the memoryless property.
Question 3
a Let X be an exponential(1) random variable. Determine the density function of U = 1/X. 2 point(s)
b Let X be a random variable with distribution function F(x) = x2 10
c Let (X, Y) be a random vector with density that is uniform on the square [0, 1] × [0, 1]. What is the distribution function of Z = max(X, Y)? 3 point(s)
d We have 4 dice at hand, throw them on the table, and see what face each one lands on. Let
ω = (ω1, ω2, ω3, ω4 ), with ωi being the face (1, 2, 3, 4, 5 or 6) die i lands on. Let Ω be the set of all outcomes ω . Define the random variables Xi(ω) := ωi, i = 1, 2, 3, 4. Assume that P is uniform on Ω .
(i) How many elements does Ω contain? 1 point(s)
(ii) Are the random variables X1 + X2(2), X3(3) + X4(4) independent? 1 point(s)
(iii) What is the probability of the event that X1 + X2 + X3 is even? 2 point(s)
Question 4
a Find the probability generating function G(s) = E(sX ) of a random variable X such that P(X = 2k) = 1/2k, k = 1, 2, . . . What is the radius of convergence of the corresponding power series? 3 point(s)
b Determine the probability generating function of X1 + X2 when X1, X2 are independent and each geometric(p)? 2 point(s)
c Determine the moment generating function M(t1, t2 ) = E[et1X1+t2X2] of the random vector
(X1 .X2) = (Z1 + Z2, Z1 − Z2), when Z1, Z2 are i.i.d. normal(0, 1) each. 3 point(s)
d What is the distribution of the sample mean of n i.i.d. normal(0, 1) random variables? 2 point(s)
2023-03-21