Stat 331/Elec 331, Midterm Exam, due October 9, at 5 pm
Solutions should be clear, complete and easy to follow. You are allowed to
to use the book, lecture notes and previous homework with solutions. Col-
laboration is not allowed. The time limit is five hours. The maximum score
is given at the end of each problem. Late turn-ins are not accepted.
1. The function fis defined as f(x) = cx2,0< x < 1 (and 0 otherwise).
a. Determine the constant cso that this becomes a pdf for a continuous
random variable X.
b. Compute E[X], V ar[X] and P(X > 0.5).
c. Let Y=√Xand find the pdf for Y.
d. Compute E[Y] and V ar[Y]. (4)
2. Are the following claims true or false? Give proofs or counterexamples.
a. If Ais an event such that P(A) = 0, then A=∅.
b. If pis the pmf for a discrete random variable then p(x)≤1 for all x.
c. If fis the pdf for a continuous random variable then f(x)≤1 for all x.
d. If X∼exp(λ), then 2X∼exp (λ/2)
e. If X∼unif (−1,1), then |X| ∼ unif (0,1).
f. If X∼bin (n, p), then 2X∼bin (2n, p). (6)
3. A transmittor sends 0’s and 1’s to a receiver. Each digit is received cor-
rectly (0 as 0, 1 as 1) with probability 0.90. Digits are received correctly
independently of each other and on the average twice as many 0’s as 1’s are
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