Deadline:
For DL Students: 15th march
For Regular Students: 10th march

Source: Textbook
Q 4-5,
Find xu for u= 0.1, 0.2 … 0.9
a) if x is uniform in the interval (0,1);
b) if f(x)= 2e-2x U(x)

Q 4-7,
Show that if the uniform variable x has an Erlang density with n=2, then Fx(x) = (1-e-cx-cxe-cx) U(x)

Q 4-8,
The random variable x is N (10; 1), Find f (x | (x-10)2 <4)

Q 4-9,
Find f(x) if F(x) = (1-e-ax) U(x-c).

Q 4-10,
If x is N (0, 2) find
a) P{1≤ x ≤ 2}
b) P{1≤ x ≤2 | x ≥ 1}

Q4-14,
A fair coin is tossed 900 times and the random variable x equals the total number of heads. a) Find fx(x), 1: exactly, 2: approximately Gamma Distribution eq. b) Find P {435 ≤ x ≤ 460}.

Q4-25,
If P (A) = 0.6 and k is the number of successes of A in n trials, a) Show that P {550 ≤ k ≤ 650} = 0.999, for n=1000.
b) Find n such that P {0.59n ≤ k ≤ 0.61n} = 0.95

Q 4-26,
A system has 100 components. The probability that a specific component will fail in the interval (a, b) equals e-a/T – e-b/T. Find the probability that in the interval (0, T/4), no more than 100 components will fail.

...Exercise
Chapter 3 Probability Distributions
1. Based on recent records, the manager of a car painting center has determined the following probability distribution for the number of customers per day. Suppose the center has the capacity to serve two customers per day.
|x |P(X = x) |
|0 |0.05 |
|1 |0.20 |
|2 |0.30 |
|3 |0.25 |
|4 |0.15 |
|5 |0.05 |
a. What is the probability that one or more customers will be turned away on a given day?
b. What is the probability that the center’s capacity will not be fully utilized on a day?
c. At least by how many, the capacity must be increased so the probability of turning a customer away is no more than 0.1?
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...

...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 that this is a valid probability...

...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...

... | Frequently | Occasionally | Not at all | Total |
Male | 221 | 456 | 795 | 1472 |
Female | 207 | 430 | 741 | 1378 |
Total | 428 | 886 | 1536 | 2850 |
A person is selected at random from the sample.
a) What is the probability the person is female or occasionally involved in charity work?
b) Are the events “being female and occasionally involved in charity work” and “being frequently involved in charity work” mutually exclusive?
yes
6. A company gave psychological tests to perspective employees. The randomvariable x represents the possible test scores.
a) Use the histogram to find the probability that a person selected at random from the survey’s sample had a test score of more than two.
b) Find the probability that the person had a test score of at most 2.
7. The following table is a frequency distribution for the number of dogs per household in a small town.
Dogs | 0 | 1 | 2 |
Households | 931 | 297 | 180 |
a) Construct the probability distribution. (round to the thousandths place)
x | 0 | 1 | 2 |
P(x) | 931/1408=0.661 | 297/1408=0.211 | 180/1408=0.128 |
b) Find the mean of the probability distribution. (round to the nearest tenth)
c) Using the mean from part b find the standard deviation of the probability distribution.
8. A computer password consists of two letters followed by a...

...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...

...THE 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...

...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...

...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 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...

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