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.

Example 24 (continued). For the exponential density, f(x) = e x,
= = 2/2 and = = 1/ so that equation (13) yields 2 = V(x) = 2 = 1/2 . (Note that the exponential pdf is the only Pearsonian statistical model with CVx = 100%.)

(3) The 3rd central moment, 3, is a measure of skewness (bear in mind that 3 0 for all symmetrical distributions). If X is continuous, then
3 = E[(X )3] =
= 3 + 2 3 (14)

For the exponential pdf , we have shown that = 1 = 1/, = 2!/ 2 and you may verify that 3 = 3! /3 = 6...

...
Event A is rolling a die and getting a 6. Suggest another event (Event B) that would be independent from Event A.
A company runs 3 servers, each providing services to 40 computers. For each server, two of its client computers are infected. What is 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...

...Week 3: Discrete RandomVariables
Stephen Bush Department of Mathematical Sciences
MM2: Statistics
- Week 3 -
1
RandomVariables
• Reference: Devore § 3.1 – 3.5 • Definitions:
• An experiment is any process of obtaining one outcome where the outcome is uncertain. • A randomvariable is a numerical variable whose value can change from one replicate of the experiment to another.
• Sample means and sample standard deviations are randomvariables
• They are different from sample to sample. • Population means and standard deviations are not random.
MM2: Statistics - Week 3 2
RandomVariables - Examples
• Experiment 1: Pick a student at random from the class
• Let X denote the height of the student
• Experiment 2: Throw a fair dice
• Let X denote the outcome of the dice. X = 1,2,3,4,5, or 6.
• Notice that the outcome of both of these events changes every time you take a new sample.
MM2: Statistics
- Week 3 -
3
1
RandomVariables
• A randomvariable can be continuous or discrete.
• Continuous randomvariables can take any real value, such as measurements. • Electrical current, length, pressure, temperature, time voltage, weight etc. • Discrete random...

...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...

...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...

...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...

...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 randomvariable
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 randomvariable 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 randomvariable 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 randomvariable
X having the density function
fx= 21-x,0<x<10,elsewhere
ﬁnd the average proﬁt per automobile.
4.38 The proportion of people who respond to a certain mail-order solicitation is a random...

...2011
Please use your calculators and give your ﬁnal answers to 3 signiﬁcant ﬁgures. Show your work for full credit. Please state clearly all assumptions made.
1. Classify each randomvariable as discrete or continuous. (a) The number of visitors to the Museum of Science in Boston on a randomly selected day. (b) The camber-angle adjustment necessary for a front-end alignment. (c) The total number of pixels in a photograph produced by a digital camera. (d) The number of days until a rose begins to wilt after it is purchased from a ﬂower shop. (e) The runnning time for the latest James Bond movie. (f) The blood alcohol level of the next person arrested for DUI in a particular county. 2. A bagel shop sells only two diﬀerent types of bagels: plain (P) and cinnamon raisin (C). Five customers are selected at random. Past records have shown that the demand for cinnamon bagels is twice that for plain bagels. Each customer buys only one bagel and the experiment consists of recording what kind of bagel these ﬁve customers buy. Let the randomvariable X be the number of people who buy a plain bagel. (a) Find the probability distribution for X. (b) Suppose at least 3 people buy a plain bagel. What is the probability that exactly 4 people buy a plain bagel? 3. The probability distribution for a discrete randomvariable X is given by the formula p(r) = for r = 1, 2, . . . , 6. (a) Verify...

...is 10%, 30%
and 50% of the initial amount. 7/8 of the coupons give the shopper 10% oﬀ, 3/32 give 30% oﬀ, and
1/32 give 50% oﬀ. Suppose that half of the customers purchase a sweater that retails for $100 and
another half purchases a suit that retails for $150. What’s the probability that a randomly selected
customer saves more than $40 by using one of these coupons?
(a) 1/8
(b) 1/16
(c) 5/64
4
(d) 3/32
(ISOM2500)[2012](f)midterm1~=0zvopee^_78631.pdf downloaded by mhwongag from http://petergao.net/ustpastpaper/down.php?course=ISOM2500&id=0 at 2013-12-16 02:44:12. Academic use within HKUST only.
18. The following plots show the probability distribution functions of four randomvariables: X, Y, Z
and W .
X
Y
Z
W
Based on these plots, which randomvariable has the largest SD? (You do not need to compute the
SD from the shown probabilities to answer this question)
(a) X
(b) Y
(c) Z
(d) W
19. A game involving chance is said to be a fair game if the expected amount won or lost is zero. Consider
the following arcade game. A player pays $1 and chooses a number from 1 to 10. A spinning wheel
then randomly selects a number from 1 to 10. If the numbers match, the player wins $x (and gets
his/her $1 back); otherwise the players loses the $1 entry fee. What value should x be so that the
game is fair?
(a) 1
(b) 5
(c) 9
(d) 10
20. A plumber loads his truck each...

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