3.1. Let X denote the number of hours Joe studies during a randomly
chosen day. Suppose the probability distribution of X has the
following form, where k is some number:
P(X=x)
=
=
=
=
0.2
kx
k(6-x)
0
if x=0
if x=1 or 2
if x=3 or 4
otherwise
a) Find the value of k.
b) Draw a graph of the probability distribution.
c) What is the probability that Joe studies at least 2 hours? exactly
2 hours? no more than 2 hours?
d) What is the conditional probability that Joe studies at least 3
hours given that he studies at least 1 hour?
a)
0.2+k+2k+3k+2k=1
8k=0.8
k=0.1
b) line chart
c)
p(x>=2)=p(2)+p(3)+p(4)
=0.2+0.3+0.2=0.7
p(x=2)=0.2
p(x<=2)=p(x=2)+p(x=1)+p(x=0)=0.2+0.1+0.2=0.5
d) A=p(x>=3) B=p(x>=1)
p(A/B) = p(A and B)/p(B)
p(A)=p(3)+p(4)=0.3+0.2=0.5
p(B)=p(1)+p(2)+p(3)+p(4)=0.1+0.2+0.3+0.2)=.8
p(A/B) = p(A and B)/p(B) = 0.5/0.8
3.3. A multiple choice test has 3 questions, each with 4 possible answers,
1 of which is correct. Let X denote the number of correct answers.
a) What values of X are possible?
b) If pure guesswork were used in taking the test, what probability
would you assign to answering the first question correctly?
c) If pure guesswork were used in taking the test, what would be
the probability distribution of X?
a) 0, 1, 2, 3
b) 1/4
c)
x
p(x)
---------0
27/64
1
27/64
2
9/64
3
1/64
3.5. Suppose a poker hand of 5 cards is to be dealt from a well-shuffied
deck of cards. Let A denote the number of aces in the hand. Find
the probability distribution of A.
A
p(A)
-------------------------------------------------0
0.658842
[35673/ 54145]
1
0.2994736
[16215/ 54145]
2
0.03992982
[ 2162/ 54145]
3
0.001736079
[94/ 54145]
4
0.00001846893
[1/ 54145]
choose(4,0)*choose(48,5)/choose(52,5)
0.658842 [divide numerator and denminator by 48 to obtain: 35673/ 54145]
choose(4,1)*choose(48,4)/choose(52,5)
0.2994736
choose(4,2)*choose(48,3)/choose(52,5)
0.03992982
choose(4,3)*choose(48,2)/choose(52,5)
0.001736079
choose(4,4)*choose(48,1)/choose(52,5)
0.00001846893
3.9. A continuous random variable X can take only values between 0
and 1. Its probability density function is p(x) = 2x.
a) Graph this density function.
b) Find the probability that X is less than 1/2.
c) Find P(1/3 < X < 2/3).
a) rectangle
b) integrate 2x from 0 to 1/2 (x^2 = 1/4)
c)P(1/3 < X < 2/3) -> (2/3)^2 - (1/3)^2 = 1/3
4.1 There are two red, three blue, and five white balls in an urn. A
ball is drawn. If it is red, the player receives $3.00; if it is blue,
$1.00; and if it is white, he pays $2.00. Let X denote his net gain.
a) What is the expected value of X?
b) What is the variance of X?
RR BBB WWWWW
a) 0.2*3+0.3*1-0.5*2=-0.1
b)E(X^2)-X-bar^2=> 0.2*(3^2)+0.3*(1^2)+0.5*((-2)^2)-((-0.1)^2)
[1] 4.09
4.2. Suppose the scores on a set of examination papers are changed by
(a) adding 10 points to each score, (b) by increasing each score by
10 percent. What effects will these changes have on the arithmetic
mean and on the standard deviation of the exam scores?
a) arithmetic mean increases by 10; sd unchanged
b) arithmetic mean increases by 10%; sd increases by 10%
4.3. A florist stocks a perishable flower that costs him 50¢ and that he
prices at $1.50. The flowers are delivered to his shop early in the
morning. Any flowers that are not sold the day of delivery are worthless
and must be thrown out. Let X be a random variable denoting
the number of these flowers that his customers order in a day. The
florist has found from past experience that the probability distribution
of X is given by
P(X
P(X
P(X
P(X
=
=
=
=
0)
1)
2)
3)
=
=
=
=
0.1
0.4
0.3
0.2.
a) Find the expected value of X. Find the variance of X.
b) If the florist orders one flower each day what is the probability
that he will sell it? If he does, what will his net profit be for the
day? If he does not, what will his net profit be for the day? What
will be the expected value of his daily net profit?
c) How many flowers should the florist stock in order to maximize
the expected net profit?
a)sum(x*p(x))=1.6, var(x)=E(x^2)-(E(x)^2)= 3.4-1.6^2=0.84
b)p(x>=1)=0.9
net profit if he sells 1 =revenue-cost=1.5-0.5=$1
if he does not sell it, his net profit = the cost = $-0.5
c)
if
if
if
order 2
orders 1, expected value of daily profit =0.9*1.5-0.5=0.85
orders 2, expected value of daily profit 1.1
orders 3, expected value of daily profit 0.9
4.4 A sample of 11 men have reported incomes of $7,800, $9,000, $7,700,
$8,400, $8,600, $9,200, $8,200, $8,600, $8,300, $7,800, and $8,800.
a) Calculate the mean, median, and sample standard deviation for
these data.
mean=8400 median=8400
sd=500
b) Suppose it is discovered that Mr. A, who reported income of
$9,200, has understated his actual income. If the correct income
figure for Mr. A is used, the mean income of the 11 men is $8,900.
What is Mr. A's income? What is the median income for the group?
sum(x)=92400
new mean=8900 incorrect value=9200
sum(x)-9200= 83200
11*8900-83200 = 14,700 (Mr A's income)
median unchanged (=8400)
4.7 Mr. Smith bought $1,000 worth of a stock when its price was $10 per
share and another $1,000 worth of the stock when its price was $40
per share. What is the average cost per share of his holdings of the
stock?
(100*10+25*40)/125 = $16
4.11 Suppose X is a random variable with mean 15 and standard deviation 3.
Suppose Y = 10 - 2X.
a) Find the mean of Y
b) Find the standard deviation of Y
c) Find E(Y^2)
a) -20
b) 6
c) 436 (=var(Y)+(meanY ^2))