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Overloading an Elevator
A new elevator in a large hotel is designed to carry about 30 people, with a total weight of up to 5000 lbs. More that 5000 lbs. overloads the elevator. The average weight of guests at the hotel is 150 lbs., with a standard deviation of 55 lbs. Suppose 30 of the hotel’s guests get into the elevator . Assuming the weights of the guests are independent random variables, what is the chance of overloading the elevator ? Give your approximate answer as a decimal.
[Pitman p 204, # 19]
Indicator Functions and Expectations – II
Let \(A\) and \(B\) be two events and let \(\mathbf{1}_A\) and \(\mathbf{1}_B\) be the associated indicator functions. Answer the following questions in terms of \(\mathbf{P}(A)\), \(\mathbf{P}(B)\), \(\mathbf{P}(B \cup A)\) and \(\mathbf{P}(B \cap A)\).
- Describe the distribution of \( \mathbf{1}_A\).
- What is \(\mathbf{E} \mathbf{1}_A\) ?
- Describe the distribution of \(\mathbf{1}_A \mathbf{1}_B\).
- What is \(\mathbf{E}(\mathbf{1}_A \mathbf{1}_B)\) ?
The indicator function of an event \(A\) is the random variable which has range \(\{0,1\}\) such that
\[ \mathbf{1}_A(x) = \begin{cases} 1 &; \text{if $x \in A$}\\ 0 &; \text{if $x \not \in A$} \end{cases}\]
2nd Moment of Shifted Random Variables
Let \(X\) be a random variable with \(\mathbf{E}(X)=\mu\) and \(\mathbf{Var}(X)=\sigma^2\). Show that for any constant \(a\)
\[\mathbf{E}\big[(X-a)^2\big]=\sigma^2+(\mu-a)^2\]
Random Digit
Let \(D_i\) be a random digit chosen uniformly from \(\{0,1,2,3,4,5,6,7,8,9\}\). Assume that each of the \(D_i\) are independent.
Let \(X_i\) be the last digit of \(D_i^2\). So if \(D_i=9\) then \(D_i^2=81\) and \(X_i=1\). Define \(\bar X_n\) by
\[\bar X_n = \frac{X_1 + \cdots+X_n}{n}\]
- Predict the value of \(\bar X_n \) when \(n\) is large.
- Find the number \(\epsilon\) such that for \(n=10,000\) the chance that you prediction is off by more than \(\epsilon\) is about 1/200.
- Find approximately the least value of \(n\) such that your prediction of \(\bar X_n\) is correct to within 0.01 with probability at least 0.99 .
- If you just had to predict the first digit of \(\bar X_{100}\), what digit should you choose to maximize your chance of being correct, and what is that chance ?
[Pitman p206, #30]
Expectation of geometric distribution
Compute the expectation of the geometric distribution using the fact that in this case
\(\mathbf{E}(X)= \sum_{k=1}^{\infty} \mathbf{Pr}(X\geq k) \)
Basic Random Walk
Consider the following “game”: A marker is placed on the real line at the point zero. On each turn a coin is flip which a 1 printed on one side and a -1 printed on the other. If the 1 side lands face up, the marker is moved on unit in the positive direction while if the -1 lands heads up then the marker is moved one unit in the negative direction. If the coin has a probability of \(p\) of landing with the 1 side face up, answer the following questions:
- Let \(p=\frac12\). After 10000 turns if you had to pick one site to find the marker which would you choose ?
- Again let \(p=\frac12\). What is the approximate chance that the marker is further then 100 units from this most likely point after 10000 turns ? What is the approximate chance that the marker is further then 300 units from this most likely point after 10000 turns ?
- Repeat the above questions with \(p=\frac{9}{10}\).
Stuffing Envelopes
You write a stack of thank you cards for people who gave you presents for your birthday. You address all of the envelopes but before you can stuff them you are called away. A friend tying to help you see the stack of cards and stuffs them in the envelops. Unfortunately they did not realize that each card was personalized and just stick them in the envelops randomly. Assuming there were \(n\) cards and \(n\) envelops, let \(X_{n}\) be the number of cards in the correct envelope.
- Find \(\mathbf{E} (X_n)\).
- Show that the variance of \(X_n\) is the same as \(\mathbf{E} (X_n)\).
- Is there any common distribution which has the above statistics ?
- (**) Show that
\[ \lim_{n\rightarrow \infty} \mathbf{P}(X_n =m) = \frac{1}{e\, m!}\]
Putting expectations together
Suppose \(\mathbf{E}(X^2)=3\), \(\mathbf{E}(Y^2)=4\) and \(\mathbf{E}(XY)=2\). What is \(\mathbf{E}[(X+Y)^2]\) ?
Expection and dice rolls
A standard 6 sided die is rolled three times.
- What is the expected value of the first roll ?
- What is the expected values of the sum of the three rolls ?
- What is the expected number of twos appearing in the three rolls ?
- What is the expected number of sixes appearing in the three rolls ?
- What is the expected number of odd numbers ?
Based on [Pitman, p. 182 #3]
Dice rolls: Explicit calculation of max/min
Let \(X_1\) and \(X_2\) be the number obtained on two rolls of a fair die. Let \(Y_1=\max(X_1,X_2)\) and \(Y_2=\min(X_1,X_2)\).
- Display the joint distribution tables for \( (X_1,X_2)\).
- Display the joint distribution tables for \( (Y_1,Y_2)\).
- Find the distribution of \(X_1X_2\).
Combination of [Pitman, p. 159 #4 and #5]
Blocks of Bernoulli Trials
In \(n+m\) independent Bernoulli \((p)\) trials, let \(S_n\) be the number of successes in the first \(n\) trials, \(T_n\) the number of successes in the last \(m\) trials.
- What is the distribution of \(S_n\) ? Why ?
- What is the distribution of \(T_m\) ? Why ?
- What is the distribution of \(S_n+T_m\) ? Why ?
- Are \(S_n\) and \(T_m\) independent ? Why ?
- Are \(S_n\) and \(T_{m+1}\) independent ? Why ?
- Are \(S_{n+1}\) and \(T_{m}\) independent ? Why ?
Based on [Pitman, p. 159, #10]
