Linear Algebra and Probability

Question 1

Albert has saved his homework in a flash drive and put the drive in a drawer, without realizing that there are already 9 drives in that drawer. Now, there are 10 drives, and they all look identical.

  • If he now picks up the drives randomly one by one and checks whether it contains his homework, what is the probability that the kth drive that he picks up contains his homework, for k = 1,2, 3, . . . , 10? Summarize your result in a table to show the probability distribution.
  • Let X be the number of drives that he picks up until he finds his homework. Find the expected value E(X).
  • Find the variance V (X).

Question 2.

 

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  • Every day Albert jogs on a circular track that is 5-kilometer long. Yesterday, he dropped a flash drive somewhere on the track. He is going back today to look for the drive.
  • If he picks a random location on the track to start, and then walks in a clockwise direction until he finds the drive, what is the probability that the he will find the drive after walking for no more than x kilometer?
  • Let X be the distance, measured in km, that he will have walked when he finds the drive. Find the expected value E(X).
  • Find the variance V (X).

Question 3.

Becky is playing a board game. She throws two fair, 6-faced dice with the numbers 1–6 on them, and the sum will determine how many spaces she will move.

  • Let X be the number of spaces that she will move. Find the expected value E(X).
  • Find the variance V (X).

Question 4.

Becky is playing a board game in which she glides along a path continuously instead of jumping from one space to another. She uses a random number generator to get two numbers from the continuous uniform distribution on [1,6], and the sum will determine how far she will move.

  • Let X be the distance that she will move. Find the expected value E(X).
  • Find the variance V (X).

Hint: The uniform distribution is on [1,6], not [0,6]. Think about the implications.

Question 5.

If we pick a number k randomly from the uniform distribution on [0, 1], what is the probability that the equation x2 + 5kx + 1 = 0 has 2 distinct real roots?

Questions 6–10 are on the next page.

Question 6.

If we pick a number k randomly from the normal distribution with µ = 1, σ = 2; what is the probability that the equation x2 + kx + 1 = 0 has 2 distinct real roots?

Question 7.

  • large group of people are raising money for charity. The amounts of money that they have raised are normally distributed, with mean being $100 and standard deviation being $30. If we select one person at random, and consider the amount of money raised by this person;

    • what is the probability that this amount exceeds $110?
    • what is the probability that this amount is less than $105?
    • what is the probability that this amount is between $85 and $100?

Question 8.

  • Write a program to generate random numbers in the following manner:

    • Generate a random number x from the continuous uniform distribution on [0,1).
    • √ • Calculate y = x.
    • Use y as the output.
  • Conduct a numerical experiment to see how the random numbers are distributed. Use the same setup as in Homework 1: provide extensive comments to the source code, submit the source code, conduct 10 experiments, with 106 trials in each experiment, and tabulate the results. To show the results:

    • If you are comfortable using numpy, matplotlib, etc, to make histograms; please do that.
    • If not, here is a quick-and-dirty way to see what the distribution may be:

      • Divide the interval [0,1) into n sub-intervals, for some large n. (For our purpose, choosing n = 10 or larger would be okay.)
      • Count how many random numbers there are in each sub-interval.
      • Scale to obtain the percentages.
      • Tabulate or plot the percentages.
  • Comment on the results. What is the probability distribution of y? Justify you claim algebraically.

Question 9.

  • There are 10 cards with the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 on them. If we randomly draw a card, with each card being equally likely to be drawn, what are the expected value and the variance?

Question 10.

  • In a standard deck of playing cards, there are 4 suits, and in each suit, there are 13 cards: A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, and K.
  • Suppose we count A as 1; and J, Q, and K as 10; and all the other cards at the numbers being shown. If we randomly draw cards from such a deck, with each card being equally likely to be drawn, what are the expected value and the variance?
  • Suppose we count A as 11; and J, Q, and K as 10; and all the other cards at the numbers being shown. If we randomly draw cards from such a deck, with each card being equally likely to be drawn, what are the expected value and the variance?

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