W12 - HW Solutions

01

(a)

.

By Markov’s Inequality, .

---

(b)

. Chebyshev Inequality: where such that .

Thus, .

---

(c)

Since is exponential:

Clearly, the actual answer is quite small compared to our previously found upper bounds.

03

(a) Let be the number of items produced in a week. We know . By Markov’s Inequality,

---

(b) By the Chebyshev Inequality,

---

(c) Similarly, by the Chebyshev Inequality:

Our inequality switches to since the Chebyshev Inequality gives an upper bound for , and we have a negative sign in front of the concerned quantity.

08

(a)

---

(b)

09

(a)

---

(b)

10

(a)

---

(b)