Probability Theory and Statistics

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Probability Theory and Statistics

1. The mean square error of the sample mean of n observations (used as an estimator for the true mean)

is:
The square root of this quantity, e, is often referred to as the standard error of the estimate.

e=E[(X−μX)2]=Var(X)= Var(X) √n

  1. (a) For the case that X is a Gaussian random variable, confirm that the sample mean will be within one standard error of μX with probability 0.68.
  2. (b) Suppose we perform a sequence of n independent Bernoulli(p) trials. How many trials are required to guarantee that X , used to estimate p, has standard error ≤ 0.05 ?

2. X and Y are joint random variables. In order to estimate the covariance of X and Y , we make n joint measurements independently; the results are (X1,Y1),(X2,Y2),…(Xn,Yn). Note that Xi and Yj are independent random variables if i ̸= j (because they come from different measurements), but they are not independent if i = j (because they come from the same measurement.) Consider the following estimator for the covariance:

C(X,Y)= 1∑(Xi−X)(Yi−Y) ni

Calculate the bias of C, which is E[C]−Cov(X,Y), the expected difference between the estimated covariance and the true covariance. Is C unbiased? If it is biased, can you fix it? A useful result: if P and Q are independent random variables, E[PQ] = E[P]E[Q].

  1. Given the following i.i.d. observations of a random variable Y : 15.2, 12.0, 14.0, 13.4, 15.0, 12.5
    1. (a) Calculate the sample mean, sample variance, and sample standard deviation.
    2. (b) Calculate an α = 0.05 confidence interval for the mean, using the Central Limit Theorem approximation.
    3. (c) Calculate an α = 0.05 confidence interval for the mean, using Student’s t-distribution. (A t-table is posted on the eLearning website.)
    4. (d) Estimate how many measurements you would need to make in order to get a confidence interval for the mean that is ±1.0 wide. Answer this using both the Central Limit Theorem and the
      t -distribution.
    5. (e) Calculate an α = 0.05 confidence interval for the variance, using the χ2 distribution. (A χ2 table is posted on the eLearning website.)
  2. Suppose that you are testing a drug which is supposed to prevent strokes. In the general population, 80-year-old people have a “first-stroke annual rate” of 1.4%; that is, P = 0.014 that a randomly selected 80-year old who hasn’t had a stroke before will have one in the coming year. You recruit 10,000 such 80-year-olds to your study. You give half the drug, and half a placebo (an inactive substitute pill, which shouldn’t have an effect.)
    1. (a) Calculate the expected number of strokes in the placebo group, and estimate a α = 0.05 confidence interval for this quantity.
    2. (b) Suppose that, after a year, 61 people in the “drug” group have had strokes, and 73 people in the “placebo” group have had strokes. Do you think the drug actually works? Why, or why not? Justify your answer by estimating the probability of a stroke in each group, along with an
      α = 0.05 confidence interval.

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