Here is the worksheet for Section 4.
Many people found these questions challenging. Here are some hints walking you through the solutions. If you solve a problem, please post your answer in the comments for others to see.
Hints/Comments:
(1) Most people seemed to figure this out. The first important conceptual challenge is realizing that the probability is simply a function, as in ![f(p) = Pr[h\ heads\ and\ t\ tails]](https://s0.wp.com/latex.php?latex=f%28p%29+%3D+Pr%5Bh%5C+heads%5C+and%5C+t%5C+tails%5D&bg=fafcff&fg=2a2a2a&s=0&c=20201002)
. Secondly, some people didn’t remember the chain rule for derivatives.
(2) The way I like to prove this is to start with the right-hand side, ![E[X] + E[Y]](https://s0.wp.com/latex.php?latex=E%5BX%5D+%2B+E%5BY%5D&bg=fafcff&fg=2a2a2a&s=0&c=20201002)
. Then, expand both using the definition of the expected value. The next crucial step is to apply the lemma I discussed in section, which is simply the marginal distribution for two variables. Finally, some manipulation of sums using the distributive property of addition will lead you to ![E[X + Y]](https://s0.wp.com/latex.php?latex=E%5BX+%2B+Y%5D&bg=fafcff&fg=2a2a2a&s=0&c=20201002)
.
(3) Many people had trouble with this. To make it simpler, assume 
for some value 
in the range of 
. Then, also assume 
and 
. Now, use the definitions of cumulative probability and expected value to arrive at the answer.
(4) First, write down (from the definition) the expected value of the random variable 
. Now, try to see how you can manipulate the terms in the sum to make it look like the desired equation. Remember that the range of 
is not necessarily the same as the range of 
.
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Foundations of CS II 
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