1st HW hints

• 1.2•Calculate it for a few, then guess the general fomula. Prove it by induction.
• 1.5•Guess the formula. You can use the result of 1.4. Prove it by induction. Use the hint on the exercise sheet about $P_n$ during the induction.
• 1.6•Use the result of 1.5.
• 1.11•Similar to 1.10. You can even do it without any summation.

2nd HW hints

• 2.2•Part a is easy. Write down and solve the equations for part b.
• 2.3•We know the formula for the sum of two independent random variables. The density does not exist necessarily!
• 2.6•Similar to 2.5. You can use the result of 2.4 and the hint on the paper.
• 2.9•Similar to 2.8.

3rd HW hints

• 3.5•Similar to 3.4.
• 3.9•Similar to 3.8 but an upgraded version. Condition, then use the hint on the paper.
• 3.12•The random variable $X\sim \text{Geo}(p)$ has probability mass function $k\mapsto p(1-p{)}^k,k\ge 0$. Then $X+1$ has optimistic geometric distribution with parameter $p$ and probability mass function $k\mapsto p(1-p{)}^{k-1},k\ge 1$.
• 3.13•Use what we know about the PGF of $\sum _{i=1}^{\nu }{X}_i$.

4th HW hints

• 4.2•Use the results of 4.1 about the joint PGF.
• 4.3•Use the formula in 3.11.
• 4.8•Use the result of 4.5.
• 4.10•Use similar ideas as in the case of the SSRW.

5th HW hints

• 5.5•Similar to 5.4.
• 5.6•Use the hint on the paper.
• 5.7•Use the hint on the paper.
• 5.8•Recall the so-called Turbo-Markov from the lecture.