Project #12482 - Probability

Homework 3
Applications of Probabilistic Models in Molecular Biology 2013 September 11

1. The American Cancer Society has calculated the annual incidence of breast cancer for women aged 50 to 54 years old to be 0.003. Breast cancer is detected from a mammography exam, which is very error prone. A study from the scientific journal JNCI has determined that 10% of the time, a mammography exam will incorrectly report that a patient has cancer when in fact they do not. It also determined that 20% of the time, the exam will incorrectly report that a patient is healthy when in fact they have cancer.

  1. (a)  What are the two “stochastic experiments” involved?

  2. (b)  Define two sample spaces: one for breast cancer and one for the mammography exam.

  3. (c)  Define a random variable called B for breast cancer presence and a random variable called M for the mammography exam result.

  4. (d)  Determine the marginal distribution for B. ????

  5. (e)  Determine the conditional distribution M ???????? B.

  6. (f)  Determine the joint distribution (M, B).

  1. (g)  Determine the marginal distribution for M.

  2. (h)  Use Bayes’ Rule to determine the posterior distribution B ???????? M.

  3. (i)  A woman undergoes a mammography exam, tests positive for breast cancer, and begins to panic. If you were a Bayesian statistician, what might you say to calm her down?

  1. Consider an “occassionally dishonest” casino that uses two kinds of dice. Of the dice, 99% are fair, but 1% are loaded so that a six comes up 50% of the time. We pick up a die from a table at random and roll it.

    (a) List the stochastic experiments.
    (b) Define the sample spaces.
    (c) Define the random variables.
    (d) What is the probability of rolling a six given that the dice is loaded? (e) What is the probability of rolling a six given that the dice is fair?

    (f) What is the joint probability of rolling a six and using a loaded die? (g) What is the joint probability of rolling a six and using a fair die?

  2. At a certain gas station 40% of the customers request regular gas, 35% request unleaded gas, and 25% re- quest premium gas. Of those customers requesting regular gas, only 30% fill their tanks. Of those customers requesting unleaded gas, 60% fill their tanks, while of those requesting premium, 50% fill their tanks.

    (a) List the stochastic experiments.
    (b) Define the sample spaces.
    (c) Define the random variables.
    (d) Determine the marginal distribution of the choice of gas.
    (e) Determine the conditional distribution of filling
    /not-filling the tank given the type of gas chosen.

    (f) Determine the joint distribution of filling/not-filling the tank and the type of gas chosen. (g) If the next customer fills the tank, what is the probability that regular gas was requested?

  3. There is a 50-50 chance that the queen carries the gene of hemophilia. If she is a carrier, then each prince has a 50-50 chance of having hemophilia, independently. If the queen is not a carrier, then the princes will not have the disease. Suppose the queen has had three princes, and none of them have hemophilia. Determine the posterior distribution that the queen carries the gene of hemophilia.

  4. For the questions below, answer True/False and provide a one sentence justification (in English).

    1. (a)  Two separate tests, A and B, are very effective at detecting whether a patient has cancer or not. Let the

      r.v. C indicate if a patient has cancer or not. Are A and B conditionally independent given C?

    2. (b)  In quesion 5a), are A and B independent?

    3. (c)  Someone rolls a red dice R and a blue dice B. Are R and B independent?

    4. (d)  You add the results of rolling dice R and B in question 5c) to get the sum T . (R, B, and T are all r.v.’s). Are R and B conditionally independent given T ?

1

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Subject Science
Due By (Pacific Time) 09/16/2013 12:00 am
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