Download Review for Midterm II

Review for Midterm II

Study Guide
1. How to identify each type of discrete random variable we've studied.
2. How to calculate probabilities, means, and variances for each type of discrete
random variable we've studied.
3. How to use a moment-generating function to find a mean and variance or to identify
a p.m.f of a r.v. X
4. Application of the Poisson distribution, and the Poisson approximation to the
binomial distribution.
5. The characteristics of a valid p.d.f. for a continuous random variable.
6. That to find the probability that a general, continuous random variable takes on
values in some interval, you need to find the area under the curve.
7. How to find the cumulative distribution function of a continuous random variable.
8. How to find a percentile, quartile, median using either a p.d.f or a c.d.f.
9. How to determine the expected value, variance, standard deviation, momentgenerating function for a general, continuous random variable.
10. How to use a moment-generating function M(t) to find the mean and variance of a
continuous random variable.
11. The characteristics (p.d.f., mean, variance, moment generating function) of a
uniform, exponential, gamma, chi-square and normal random variables.
12. Know how to find probabilities for the named continuous distributions we studied.
13. How to find the distribution of a function of a random variable using either the
distribution function technique or the change of variable technique.
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Hints on identifying the type of discrete random variable
In most cases, you can identify a discrete random variable through the definition of X.
1. If X is binomial, X is the number of successes selected from a finite number of
objects ("sampling with replacement"). The critical thing is to check to make sure
the 4 conditions are met. Basically requires Bernoulli trials. Remember that if the
population is large relative to the sample, you can treat X as binomial even if
sampling occurs without replacement.
2. If X is geometric, then X is the number of attempts to get 1 success. Requires
Bernoulli trials.
3. If X is negative binomial, then X is the number of attempts to get r successes.
Requires Bernoulli trials.
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Examples
1. Suppose that 25% of students at a particular college have SAT scores over 1200. If
I pick 5 students at random from this student body, what is the probability that at
least two of those picked have SAT scores over 1200?
2. Find the expected value and variance of the number of times one must throw a die
until the outcome 1 has occurred 4 times.
3. Suppose that the moment generating function of a random variable X is given
t
by M (t )  e 3(e 1) . What is P(X>1)?
4. Suppose that the moment generating function of a random variable X is given
by M (t ) 
1
,
(1  2t ) 6
2
for t  1 . If Y  2 X  4 , what is the mean of Y? (Hint:
2
First find the expected value and variance of X.)
5. Suppose that the length of a phone cell in minutes is an exponential random
variable with parameter θ=10. If someone arrives immediately ahead of you at a
public telephone booth, find the probability that you will have to wait.
a) between 10 and 20
b) more than 20 minutes given you have already waited 10minutes.
6. Suppose that the cumulative density function of a random variable X is
F ( x)  x 2 c for 0  x  2 .
a) Find c such that f(x) satisfies the conditions of a p.d.f.
b) Find the median or 50th percentile of X (e.g.,  0.5 )
c) Find the moment generating function M(t) of X.
(Hint: First find the probability of density function f(x), then use it to
compute M(t)).
7. Suppose that the random variable X has the p.d.f f ( x)  3x 2 8 , 0  x  2. Consider a
new random variable Y, where Y=X3. Find the p.d.f of Y.