Tag Archives: JCM_math230_HW6_S15

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]

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

\[\bar X_n = \frac{X_1 + \cdots+X_n}{n}\]

  1. Predict the value of \(\bar X_n \) when \(n\) is large.
  2. 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.
  3. 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 .
  4. 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]

Games with Black and White Balls

Consider the following gambling game for two players, Black and White. Black puts \(b\) black balls and White puts \(w\) white balls in a box. Black and White take turns at drawing randomly from the box, with replacement between draws until either Black wins by drawing a black ball or White wins by drawing a white ball. Suppose Black gets to draw first.

  1. Calculate \(\mathbf{P}(\text{Black wins})\) and \(\mathbf{P}(\text{White wins})\) in terms of \(p=b/(b+w)\).
  2. What value of \(p\) makes the game fair (equal chances of wining) ?
  3. Is the game ever fair ?
  4. What is the least total number of balls in the game, \((b+w)\), such that neither player has more that that \(51\%\) chance of winning ?

 

[Pitman P219, #13]

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:

  1. Let \(p=\frac12\). After 10000 turns if you had to pick one site to find the marker which would you choose ?
  2. 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 ?
  3. 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.

  1. Find \(\mathbf{E} (X_n)\).
  2. Show that the variance of \(X_n\)  is the same as  \(\mathbf{E} (X_n)\).
  3. Is there any common distribution which has the above statistics ?
  4. (**) Show that
    \[ \lim_{n\rightarrow \infty} \mathbf{P}(X_n =m) = \frac{1}{e\, m!}\]