Probability Theory and Statistics

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Probability Theory and Statistics

  1. Prove that, if X is Gaussian(μ, σ) and Y = aX + b, then Y is Gaussian(aμ + b, aσ).
  2. The peak temperature T , in Fahrenheit, on a July day in Antarctica is a Gaussian random variable with σ2 = 225 and mean μ = −75. Calculate…
    1. (a) P(T > 0)
    2. (b) P(T < −100)
    3. (c) P(−90 ≤ T < −60)
    4. (d) P(T > −75|T > −90)
    5. (e) The probability that there is at least one day in July with temperature above 0 (assuming that the peak temperatures on different days are independent.)
  3. Suppose X is Gaussian(μ,σ).
    1. (a) Find the probability of a positive “4σ event”, that is, observing X > μ + 4σ. Use the complementary CDF table posted on eLearning.
    2. (b) Suppose you perform measurements on X. If you perform 1000 measurements, what is the expected number of 4σ events? What is the probability of seeing at least one such event?
    3. (c) In a series of independent measurements of X, what is the expected number of trials required to accumulate a total of 10 4σ events?
    4. (d) Derive the conditional probability that a “4σ event”, observing X > μ + 4σ, is actually a 4.5σ event, X > μ + 4.5σ. Explain this results in terms of the shape of the “tails” of fX .
  4. To win a prize at the fair, you need to hit the bullseye once with a bow and arrow. Your probability of success is only p = 0.2 per shot. You get three shots; if you hit the bullseye, you stop shooting. Let X be the number of shots taken, and Y indicate winning the prize (Y = 1 for success, Y = 0 for failure.)
    1. (a) Calculate the joint PMF of X and Y . Report your result as a table, and confirm that your joint PMF is normalized. A tree diagram may be useful in working out the joint PMF.
    2. (b) Calculate the marginal PMFs of X and Y .
    3. (c) Calculate ρ(X,Y) and interpret its sign and magnitude. (Note: ρ(X,Y) and ρX,Y are two different notations for the same thing, the correlation coefficient.)
  5. 40% of students own an iPhone, 20% of students own an Apple computer, and 16% of students own both. Let X be a Bernoulli random variable that describes “owning an iPhone.”, and let Y be a Bernoulli random variable that describes “owning an Apple computer.”
    1. (a) Determine the joint PMF of X and Y .
    2. (b) Calculate ρ(X,Y)
    3. (c) Calculate the conditional PMF PX|B(x), where B = {Y = 1}, the even that a student owns an Apple computer.

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