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 random variable as discrete or continuous. (a) The number of visitors to the Museum of Science in Boston on a randomly selected day. (b) The camberangle adjustment necessary for a frontend 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 random variable 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 random variable X is given by the formula p(r) = for r = 1, 2, . . . , 6. (a) Verify that this is a valid probability distribution. (b) Find P (X = 4). (c) Find P (X > 2). (d) Find the probability that X takes on the value 3 or 4. (e) Construct the corresponding probability histogram. 4. Two packages are independently shipped from Fort Collins, Colorado, to the same address in Seattle, Washington, and each is guaranteed to arrive within 4 days. The probability that a package arrives within 1 day is 0.10, within 2 days is 0.15, within 3 days is 0.25, and on the fourth day is 0.50. Let the random variable X be the total...
...The mean of a discrete randomvariable X is a weighted average of the possible values that the randomvariable can take. Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a randomvariable weights each outcome xi according to its probability, pi. The mean also of a randomvariable provides the longrun average of the variable, or the expected average outcome over many observations.The common symbol for the mean (also known as the expected value of X) is , formally defined by
Variance  The variance of a discrete randomvariable X measures the spread, or variability, of the distribution, and is defined by
The standard deviation is the square root of the variance.
Expectation  The expected value (or mean) of X, where X is a discrete randomvariable, is a weighted average of the possible values that X can take, each value being weighted according to the probability of that event occurring. The expected value of X is usually written as E(X) or m.
E(X) = S x P(X = x)
So the expected value is the sum of: [(each of the possible outcomes) × (the probability of the outcome occurring)].In more concrete terms, the expectation is what you would expect the outcome of an experiment...
...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 table with all possibilities for number of occurrences of in 9 trials and calculate each’s corresponding...
...Exercise
Chapter 3 ProbabilityDistributions
1. Based on recent records, the manager of a car painting center has determined the following probabilitydistribution 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 probabilitydistribution 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...
...Probabilitydistribution
Definition with example:
The total set of all the probabilities of a randomvariable to attain all the possible values. Let me give an example. We toss a coin 3 times and try to find what the probability of obtaining head is? Here the event of getting head is known as the randomvariable. Now what are the possible values of the randomvariable, i.e. what is the possible number of times that head might occur? It is 0 (head never occurs), 1 (head occurs once out of 2 tosses), and 2 (head occurs both the times the coin is tossed). Hence the randomvariable is “getting head” and its values are 0, 1, 2. now probabilitydistribution is the probabilities of all these values. The probability of getting 0 heads is 0.25, the probability of getting 1 head is 0.5, and probability of getting 2 heads is 0.25.
There is a very important point over here. In the above example, the randomvariable had 3 values namely 0, 1, and 2. These are discrete values. It might happen in 1 certain example that 1 randomvariable assumes 1 continuous range of values between x to y. In that case also we can find the probabilitydistribution of...
...Important Discrete
ProbabilityDistributions
51
Chapter Goals
After completing this chapter, you should be able
to:
Interpret the mean and standard deviation for a
discrete probabilitydistribution
Explain covariance and its application in finance
Use the binomial probabilitydistribution to find
probabilities
Describe when to apply the binomial distribution
Use Poisson discrete probabilitydistributions to
find probabilities
52
Definitions
RandomVariables
A randomvariable represents a possible
numerical value from an uncertain event.
Discrete randomvariables produce outcomes
that come from a counting process (e.g. number
of courses you are taking this semester).
Continuous randomvariables produce outcomes
that come from a measurement (e.g. your annual
salary, or your weight).
53
Definitions
RandomVariablesRandomVariables
Ch. 5
Discrete
RandomVariable
Continuous
RandomVariable
Ch. 6
54
Discrete RandomVariables
Can only assume a countable number of values
Examples:
Roll a die twice
Let X be the number of times 4 comes up
(then X...
...Tutorial on Discrete ProbabilityDistributions
Tutorial on discrete probabilitydistributions with examples and detailed solutions.

Top of Form

Web  www.analyzemath.com 

Bottom of Form 
 Let X be a randomvariable that takes the numerical values X1, X2, ..., Xn with probablities p(X1), p(X2), ..., p(Xn) respectively. A discreteprobabilitydistribution consists of the values of the randomvariable X and their corresponding probabilities P(X).
The probabilities P(X) are such that ∑ P(X) = 1Example 1:Let the randomvariable X represents the number of boys in a family.
a) Construct the probabilitydistribution for a family of two children.
b) Find the mean and standard deviation of X.Solution to Example 1: * a) We first construct a tree diagram to represent all possible distributions of boys and girls in the family. * Assuming that all the above possibilities are equally likely, the probabilities are:
P(X=2) = P(BB) = 1 / 4
P(X=1) = P(BG) + P(GB) = 1 / 4 + 1 / 4 = 1 / 2
P(X=0) = P(GG) = 1 / 4 * The discrete probabilitydistribution of X is given by X  P(X) 
0  1 / 4 
1  1 / 2 
2 ...
...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...
...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...
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