MTH136/STA114: Statistics
Homework #1
Due: Wednesday, Jan 24, 2001
-
In a small clinical trial there are 7 successes and 3 failures among the
ten subjects. The true success probability p is unknown, and we
would like to learn more about it.
- Give the likelihood function f(p)=cP[X=7|p] for
the success probability p, normalized to be a
probability density function (as a function of p on
the interval 0<p<1), so that the integral
Int{0<p<1} f(p) = 1.
Your f(p) will be the product of a constant, a power of
p, and a power of (1-p); you must give the values
of the constant and the powers. Use Mathematica or Maple to
evaluate the required integral; in your solution, give the
Mathematica or Maple expression you used to evaluate it.
- Plot f(p) as a function of p, using your choice
of S-Plus, Matlab, Mathematica, or Maple. Turn in the plot
and the computer code used to generate it. Identify (both
numerically and graphically) the point where f(p)
attains its maximum value.
- Give the name of this distribution and the values of any
parameters. You may find this list of probability
distributions (ps, pdf) useful.
- Give the mean and standard deviation of this
distribution. Again, the distribution list may be helpful: you
are not asked to derive the mean or variance (no integration
is necessary). Remember that Gamma(N) is (N-1) factorial,
(N-1)! = 1·2·...·(N-1), for integers N.
- Find the probabilities Pr[p>0.50], Pr[p>0.90], and
Pr[p<0.01], all correct to two decimal places, using S-Plus.
Explain in your own words what are the meanings of these
probabilities, in the context of the clinical trial.
- DeGroot Chapter 3 Section 6 Page 141 Problems 8,9
- DeGroot Chapter 3 Section 7 Page 149 Problems 1,5
- DeGroot Chapter 3 Section 8 Page 158 Problem 7
- DeGroot Chapter 3 Section 9 Page 169 Problem 1,5