Instructions
This problem set is due on Friday November 29, 2019 at 23:59. Solutions to Problems should be submitted on Gradescope and will be graded. Solutions to Exercises will not be graded and you should not submit them, although we will be happy to discuss them with you.
Problems
- Prove that if $X:\Omega \to \mathbb R$ is constant (i.e. for all $\omega \in \Omega$, $X(\omega) = c$ for some $c \in \mathbb R$), then $E(X) = c$.
- Prove that $V(X) = 0$ if and only if $X$ is constant.
- The $GC$-content of a DNA string is the proportion of the string that is either a $G$ or $C$. More specifically, given a DNA string $w$, we define
$$\mathbb{GC}(w) = \frac{|w|_G + |w|_C}{|w|}.$$
Suppose we generate a random DNA string with the following probabilities:
$$\begin{matrix} \Pr(A) = 0.4 & \Pr(C) = 0.1 & \Pr(G) = 0.3 & \Pr(T) = 0.2 \end{matrix}$$
- What is the expected $GC$-content of the string?
- Use Chebyshev's inequality to give a lower bound on the probability that the number of $G$s and $C$s in a DNA string generated randomly according to the above probabilities of length 10000 does not deviate from the expected number of $G$s and $C$s by more than 100.
- Recall our noisy channel from the previous problem set, which has a $\frac 1 4$ chance of introducing an error (i.e. it sends the wrong bit). Suppose we send a stream of random bits through the channel with probability $\frac{7}{10}$ of sending a 0 and probability $\frac{3}{10}$ of sending a 1.
- What is the probability that a 0 is received?
- What is the probability that a 0 was sent, given that a 0 is received?
Exercises
- Prove that if $X$ and $Y$ are independent random variables, then
$$E(X \cdot Y) = E(X) \cdot E(Y).$$
- Prove that
$$\min_{\omega \in \Omega} X(\omega) \leq E(X) \leq \max_{\omega \in \Omega} X(\omega).$$
- Prove that if $c$ is a constant, then $V(X) = V(X+c)$.
- Prove that if $c$ is a constant, then $V(cX) = cV(X)$.
- Rosen 7.3 Exercises 9, 11, 13, 19, 21, 23
- Rosen 7.4 Exercises 19, 25, 27, 35, 39, 45