4.6 An attendant at a car wash is paid according to the number of cars that pass through. Suppose the probabilities are 1/12, 1/12, 1/4, 1/4, 1/6, and 1/6, respectively, that the attendant receives $7, $9, $11, $13, $15, or $17 between 4:00 P. M. and 5:00 P. M. on any sunny Friday. Find the attendant’s expected earnings for this particular period.

4.7 By investing in a particular stock, a person can make a proﬁt in one year of $4000 with probability 0.3 or take a loss of $1000 with probability 0.7. What is this person’s expected gain?

4.10 Two tire-quality experts examine stacks of tires and assign a quality rating to each tire on a 3-point scale. Let X denote the rating given by expert A and Y denote the rating given by B. The following table gives the joint distribution for X and Y .

4.12 If a dealer’s proﬁt, in units of $5000, on a new automobile can be looked upon as a random variable X having the density function

fx= 21-x,0<x<10,elsewhere

ﬁnd the average proﬁt per automobile.

4.14 Find the proportion X of individuals who can be expected to respond to a certain mail-order solicitation if X has the density function.

fx= 2(x+2)50<x<10,elsewhere

4.28 Consider the information in Exercise 3.28 on page 93. The problem deals with the weight in ounces of the product in a cereal box, with
fx= 25,23.75 ≤x ≤26.250,elsewhere.

4.33 Use Deﬁnition 4.3 on page 120 to ﬁnd the variance of the random variable X of Exercise 4.7 on page 117.
4.7 By investing in a particular stock, a person can make a proﬁt in one year of $4000 with probability 0.3 or take a loss of $1000 with probability 0.7. What is this person’s expected gain?

4.37 A dealer’s proﬁt, in units of $5000, on a new automobile is a random variable X having the density function given in Exercise 4.12 on page 117. Find the variance of X. 4.12 If a dealer’s proﬁt, in units of $5000, on a new automobile can be looked upon as a random variable X having the density function...

...
2. The following is the probability distribution function of the number of complaints a customer manager has to handle in half an hour.
Suppose he can handle at most 3 complaints in half an hour.
a. What is k?
b. What is the probability there are less than 2 complaints in half an hour?
c. What is the probability there are less than 2 complaints in an hour?
3. A randomvariable [pic] can be assumed to have five values: 0, 1, 2, 3, and 4. A portion of the probability distribution is shown here:
|x |0 |1 |2 |3 |4 |
|P(X = x) |0.1 |0.3 |0.3 |a |0.1 |
a. Find a
b. Find [pic], E(X2), [pic] and the standard deviation of [pic].
4. The probabilities that a building inspector will observe 0, 1, 2, 3, 4, or 5 (X) violations of the building code in a home built in a large development are given in the following table:
|x |0 |1 |2 |3 |4 |5 |
|P(X = x) |0.41 |0.22 |0.17 |0.13 |0.05 |0.02 |
Find the mean and standard deviation of the distribution.
5. The tables below are the probability distribution...

...SIDS31081 - Statistics Refresher
2006 – 2007
Exercises
(Probability and RandomVariables)
Exercise 1
Suppose that we have a sample space with five equally likely experimental outcomes :
E1,E2,E3,E4,E5.
Let
A = {E1,E2}
B = {E3,E4}
C = {E2,E3,E5}
a. Find P(A), P(B), P(C).
b. Find P(A U B) . Are A and B mutually exclusive?
c. Find Ac, Bc, P(Ac), P(Bc).
d. Find A U Bc and P(A U Bc)
e. Find P(B U C)
Exercise 2
A committee with two members is to be selected from a collection of 30 people, of whom 10
are males and 20 are females.
a. Find the probability that both members are male
b. Find the probability that both members are female
c. Find the probability that one member is male and one is female.
Exercise 3
A warehouse contains 100 tires, of which 5 are defective.
Four tires are chosen at random for a new car.
Find the probability that all four are good.
Exercise 4
In a particular city,
40% of the people subscribe to magazine A, 30% of the people subscribe to magazine B and
50% to magazine C.
However, 10% subscribe to both A and B, 25% subscribe to both A and C, 15% subscribe to
both B and C. Finally, 5% subscribe to all three magazines.
A person is chosen at random.
a. What is the probability that the chosen person subscribes to at least one magazine?
b. What is the probability that the chosen person subscribes to at least two magazines?
c. Find the conditional probability that...

...Dynise Adams
STA
Individual Work unit-8
Section 6.1
8. a) The time it takes for a light bulb to burn out is a continuous randomvariable because the time is being measured. All possible results for the variable time (t) would be greater than > 0.
b) The weight of a T-bone steak is a continuous randomvariable because the weight of the steak is measured. All the possible results for the weight of the T-bone steak would be positive numbers making the variable weight (w) > greater than 0.
c) The number of free throw attempts before the first shot is made is a discrete randomvariable because every shot is attempt can be counted. Let (x) represent shot attempts, all the possible results of the value x would be x = 0, 1, 2, 3, 4
d) In a random sample of 20 people the number with type A blood is a discrete randomvariable because the people with type A blood are being counted. Let (x) represent people with Type A blood, all possible results of the value x would be x = 0, 1, 2
12. les; because Px=1 and 0≤Px ≤1 for all x.
16. No, because P x=1.25 ≠1.
20. a) This is a discrete probability distribution because the sum of the probabilities is 1 and the probabilities are between 0 and 1.
c) mx = x ∙Px=0 0.073+10.117+20.258+30.322+40.230=2.519=2.5. Or average the...

...probability that 3 randomly chosen client computers serviced by different servers (one per server) will all be infected?
The probability that Alice’s RSA signature on a document is forged is () What is the probability that out of 4 messages sent by Alice to Bob at least one is not forged?
Event A is selecting a “red” card from a standard deck at random. Suggest another event (Event B) that is compatible with Event A.
What is the probability of getting 6 tails in 10 trials of tossing a coin? Solve this problem by using :The approximation mentioned in Theorem 6
The Binomial Distribution
Then compare answers for a) and b) after you have solved the problem.
When transmitting messages from a point A to a point B, out of every 40 messages 6 need to be corrected by applying error correcting codes. What is the probability that in a batch of 200 messages sent from A to B, there will be between 38 and 42 messages that will have to be corrected? Please choose the appropriate method to approximate this quantity.
The probability of an event occurring in each of a series of independent trials is . Find the distribution function of the number of occurrences of in 9 trials. That is, provide a table with all possibilities for number of occurrences of in 9 trials and calculate each’s corresponding probabilities.
The probability that a network will be shut down on any given day is 0.0003. What is the probability that the network will be...

...MOMENTS OF A RANDOMVARIABLE
Definition: Let X be a rv with the range space Rx and let c be any known constant. Then the kth moment of X about the constant c is defined as
Mk (X) = E[ (X c)k ]. (12)
In the field of statistics only 2 values of c are of interest: c = 0 and c = . Moments about c = 0 are called origin moments and are denoted by k, i.e., k = E(Xk ), where c = 0 has been inserted into equation (12). Moments about the population mean, , are called central moments and are denoted by k, i.e, k = E[ (X )k ], where c = has been inserted into (12).
STATISTICAL INTERPRETATION OF MOMENTS
By definition of the kth origin moment, we have:
k =
(1) Whether X is discrete or continuous, 1 = E(X) = , i.e., the 1st origin moment is simply the population mean (i.e., 1 measures central tendency).
(2) Since the population variance, 2, is the weighted average of
deviations from the mean squared over all elements of Rx, then 2 =
E[(X )2] = 2. Therefore, the 2nd central moment, 2 = 2, is a measure of dispersion (or variation, or spread) of the population. Further, the 2nd central moment can be expressed in terms of origin moments using the binomial expansion of (X )2, as shown below.
2 = E[ (X )2] = E[(X2 2 X + 2 )] = E(X2) 2 E(X) + 2
= E(X2) 2 = ()2 = 2 . (13)
Example 24 (continued). For the...

...over
2206-7303
0516
–6–
5.
M06/5/MATME/SP1/ENG/TZ2/XX
The probability distribution of the discrete randomvariable X is given by the following table.
x
1
2
3
4
5
P ( X = x)
0.4
p
0.2
0.07
0.02
(a)
Find the value of p.
(b)
Calculate the expected value of X.
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2206-7303
0616
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6.
M06/5/MATME/SP1/ENG/TZ2/XX
The graph of a function g is given in the diagram below.
The gradient of the curve has...

...complaints can be represented as a table or a graph, both shown below. The randomvariable xi represents the number of complaints, and p(xi) is the probability of receiving xi complaints.
xi
0
1
2
3
4
5
6
p(xi)
.15
.1
.28
.20
.10
.10
.07
What is the expected number of complaints received per week? Round your answer to two places after the decimal.
Answer
Selected Answer: 2.58
. Question 6 .5 out of 5 points
Deterministic techniques assume that no uncertainty exists in model parameters.
Answer
Selected Answer: True
. Question 7 .5 out of 5 points
Excel can be used to simulate systems that are represented by both discrete or continuous randomvariables.
Answer
Selected Answer: True
. Question 8 .0 out of 5 points
A cumulative probability distribution is often used for simulating the values of a discrete randomvariable.
Answer
Selected Answer: False
. Question 9 .5 out of 5 points
A time series may exhibit a trend and a cyclic behavior at the same time.
Answer
Selected Answer: True
. Question 10 .5 out of 5 points
The exponential smoothing method is particularly used for time series that exhibit only a trend behavior.
Answer
Selected Answer: False
. Question 11 .5 out of 5 points
The number of traffic accidents in a cosmopolitan area during next July is an example of a continuous...

...Discrete RandomVariables: Homework
Exercise 1
Complete the PDF and answer the questions.
|X |P(X = x) |X(P(X = x) |
|0 |0.3 | |
|1 |0.2 | |
|2 | | |
|3 |0.4 | |
a. Find the probability that X = 2.
b. Find the expected value.
Exercise 2
Suppose that you are offered the following “deal.” You roll a die. If you roll a 6, you win $10. If you roll a 4 or 5, you win $5. If you roll a 1, 2, or 3, you pay $6.
a. What are you ultimately interested in here (the value of the roll or the money you win)?
b. In words, define the RandomVariable X.
c. List the values that X may take on.
d. Construct a PDF.
e. Over the long run of playing this game, what are your expected average winnings per game?
f. Based on numerical values, should you take the deal?
g. Explain your decision in (f) in complete sentences.
Exercise 3
A venture capitalist, willing to invest $1,000,000, has three investments to choose from. The first investment, a software company, has a 10% chance of returning $5,000,000 profit, a 30% chance of returning $1,000,000 profit, and a 60% chance of losing the million dollars. The second company, a hardware company, has a 20% chance of returning $3,000,000 profit, a 40% chance of...

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