Dr. Özlem İLK
Fall 2011-2012
IAM 530
HOMEWORK 4
Due 17 November 2011, Thurday, 9:40
You should work on these questions on your own. Please feel free to get help from me or from
Asena, but not from anyone else. Please note that deadline for this homework is extremely
strict. The solutions will be provided at 12:30 on the same day, so that you can see them
before your exam.
1. You can use any method you want to solve the following questions:
a) If X~Unif(0,1), then find the pdf of Y=1-X, and state which special distribution it corresponds
to.
b) If Xi (i=1,2,...,n) form a random sample such that each Xi comes from Gamma(α,β), then find
n
 xi
the pdf of X  i 1
n
, and state which special distribution it corresponds to.
2. Suppose that X i ~ N ( ,  2 ) i=1,2,…,n and Zi ~ N(0,1) i=1,2,…,k and all variables are
independent. Find (do not just state) the exact distributions (not limiting distribution) of each
of the following variables. You can use theorems (or facts) that were discussed in class
without proving them, but you need to explicitly write down the theorem (or fact) you are
using.
a)
b)
Z1
Z 22
Z12
Z 22
1
k
c)
X
2
 zi

i 1
k
3. Mensa (from the Latin word for “mind”) is an international society devoted to intellectual
pursuits. Any person who has an IQ in the upper 2% of the general population is eligible to
join. If we assume that IQs are normally distributed with μ = 100 and σ = 16, what is the
lowest IQ that will qualify a person for membership?
4. Suppose that numerical grades in a statistics class are values of a r.v. X which is Normally
distributed with mean μ = 65 and standard deviation 15. Suppose that letter grades are
assigned according to the following rule: student receives an A if X ≥ 85; B if 70 ≤ X < 85; C
if 55 ≤ X < 70; D if 45 ≤ X < 55; and F if X ≤ 45. If a student is chosen at random from this
class, calculate the probability that the student will earn i) A; ii) B; iii) F.
5. Let f1(y) and f2(y) be density functions, and let a be a constant such that 0≤ a ≤1. Consider the
function f(y)=a f1(y) + (1-a) f2(y). Such a density function is often called a mixture
distribution. Here are some examples of real life applications for such distributions.
Example 1: Financial returns often behave differently in normal situations and during crisis
times. A mixture of two normal distributions with different means and variances can be
assumed for returns, one for the returns during normal situations, and another for during
crises.
Example 2: The prices of houses in a particular neighborhood will tend to be similar, while
prices in a different neighborhood may be extremely different. For instance, if we are to
collect prices in both Cukurambar and Sincan, we will need a mixture of two distributions to
describe these prices.
a) Show that f(y) is a valid density function.
b) Suppose that Y1 is a random variable with density function f1(y), and E(Y1)=μ1 and
Var(Y1)=  12 . Similarly, suppose that Y2 is a random variable with density function f2(y), and
E(Y2)=μ2 and Var(Y2)=  22 . Assume that Y is random variable whose density is a mixture of
the densities corresponding to Y1 and Y2.
i) Show that E(Y)= a μ1 + (1-a) μ2
2
ii) Show that Var(Y)= a  12 + (1-a)  22 + a(1-a) (μ1 - μ2)2
Hints: 1. For simplicity, you can assume that both f1(y) and f2(y) are continuous.
2. Use the fact (without proving): E(Yi2) = μi2+  i2 for i=1, 2
c) Assume that the prices in Cukurambar follow a distribution of N(400K,20K), while the
ones in Sincan follow N(80K,5K) where K stands for thousand TL.
i) Draw a realistic but not detailed histogram of these two distributions on the same picture at
hand. (You should be careful about means and variances and shape of the distribution, but
you don’t need to be rigorous about the scale.)
ii) Suppose that you would like these two neighborhoods to have equal weights in
representing Ankara. Write down the p.d.f. of this mixture distribution. Calculate the mean
and variance of the mixture distribution.
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