Tag Archives: JCM_math230_HW5_S13

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 \(\mathbf{P}(B \cap A)\).

  1. Describe the distribution of \( \mathbf{1}_A\).
  2. What is \(\mathbf{E} \mathbf{1}_A\) ?
  3. Describe the distribution of \(\mathbf{1}_A \mathbf{1}_B\).
  4. 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

\[ \mathbf{1}_A(x) = \begin{cases} 1 &; \text{if $x \in A$}\\ 0 &; \text{if $x \not \in A$} \end{cases}\]

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]

A simple mean calculation

Suppose that \(X \in \{1,2,3\}\) and \(Y = X+ 1\), and \(\mathbf{P}(X = 1) = 0.3, \ \mathbf{P}(X = 2) = 0.5,\ \mathbf{P}(X = 3) = 0.2.\)

(a) Find \(\mathbf{E}(X)\).

(b) Find \(\mathbf{E}(Y)\).

(c) Find \(\mathbf{E}(X + Y)\).

[Author Mark Huber. Licensed under Creative Commons.]

Expected Value and Mean Error

Let \(X\) be a random variable with \(\mu_1=\mathbf{E}(X)\) and \(\mu_2=\mathbf{E}(X^2)\). For any number \(a\) define the mean squared error

\[J(a)=\mathbf{E}\big[(X-a)^2\big] \]

and the absolute error

\[K(a)=\mathbf{E}\big[|X-a|\big] \]

  1. Write \(J(a)\) in terms of  \(a\), \(\mu_1\), and \(\mu_2\) ?
  2. Use the above answer to calculate \(\frac{d J(a)}{d\, a}\) .
  3. Find the \(a\) which is the  solution to \(\frac{d J(a)}{d\, a}=0 ?\) Comment on this answer in light of the name  “Expected Value” and argue that it is actually a minimum.
  4. Assume that \(X\) only takes values \(\{x_1,x_2,\dots,x_n\}\).  Use the fact that
    \[ \frac{d\ }{d a} |x-a| = \begin{cases} -1 & \text{if \(a < x\)}\\
    1 & \text{if \(a > x\)}\end{cases}
    \]
    to show that as long as \(a \not\in \{x_1,x_2,\dots,x_n\}\) one has
    \[ \frac{d K(a)}{d\, a} =\mathbf{P}(X<a) – \mathbf{P}(X>a)\]
  5. Now show that if \( a \in (x_k,x_{k+1})\) then \(\mathbf{P}(X<a) – \mathbf{P}(X>a) = 2\mathbf{P}(X \leq x_k) – 1\).
  6. The median is any point \(a\) so that both  \(\mathbf{P}(X\leq a) \geq \frac12 \) and \(\mathbf{P}(X\geq a) \geq\frac12\). Give an example where the median is not unique. (That is to say there is more than one such \(a\).
  7. Use the above calculations  to show that if \(a\) is any median (not equal to one of the \(x_k\)), then it solves  \(\frac{d K(a)}{d\, a} =0\) and that it is a minimizer.

 

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
    \[ \lim_{n\rightarrow \infty} \mathbf{P}(X_n =m) = \frac{1}{e\, m!}\]

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.

  1. What is the expected value of the first roll ?
  2. What is the expected values of the sum of the three rolls ?
  3. What is the expected number of twos appearing in the three rolls ?
  4. What is the expected number of sixes appearing in the three rolls ?
  5. What is the expected number of odd numbers ?

Based on [Pitman, p. 182 #3]