Discrete and Continuous Probability
All probability distributions can be categorized as discrete probability distributions or as continuous probability distributions (stattrek.com). A random variable is represented by “x” and it is the result of the discrete or continuous probability. A discrete probability is a random variable that can either be a finite or infinite of countable numbers. For example, the number of people who are online at the same time taking a statistics class at CTU on a given day is a discrete random probability. Another example of a discrete random probability is the number of people who stand in a checkout lane in Kroger on a given day. A continuous probability is a random variable that is infinite and the number is uncountable. An example of a continuous probability is the wait time in a Kroger line on a given day and time and that number could be 5 minutes, 5.2 minutes, or 5.34968...minutes. The same can be said if the example was the amount of silk a silk worm produced on a given day. The dice experiment is a discrete random probability because it yielded 6 possible outcomes which are 1, 2, 3, 4, 5, and 6. The number that the die landed on after each roll is “x” or the random variable. The discrete random probability is a countable number because the dice only has 6 sides. The experiment produced three 1’s, four 2’s, five 3’s, two 4’s, four 5’s and two 6’s. The experiment is a probability distribution because all six sides had the same “chance” to land on its side under the “set” conditions of 20 rolls. This is not a binomial probability because it has more than two possible outcomes. The outcome of the die has 6 possibilities or a 1 out of 6 chance to land on its side which would disqualify it as binomial. A binomial probability only deals with successes or failures. This type of experiment either does something or it doesn’t, there is no in between. References
...The mean of a discrete random variable X is a weighted average of the possible values that the random variable can take. Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a random variable weights each outcome xi according to its probability, pi. The mean also of a random variable 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 random variable 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 random variable, 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 to be on average.
2. Define the following;
a) Binomial Distribution  is the discreteprobability...
...Tutorial on DiscreteProbabilityDistributions
Tutorial on discreteprobabilitydistributions with examples and detailed solutions.

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 Let X be a random variable that takes the numerical values X1, X2, ..., Xn with probablities p(X1), p(X2), ..., p(Xn) respectively. Adiscreteprobabilitydistribution consists of the values of the random variable X and their corresponding probabilities P(X).
The probabilities P(X) are such that ∑ P(X) = 1Example 1:Let the random variable 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 discreteprobabilitydistribution of X is given by X  P(X) 
0  1 / 4 
1  1 / 2 
2  1 / 4 
* * Note...
...TEM1116 Probability and Statistics
Tri1 2013/14
Chapter 1
Chapter 1: Discrete and ContinuousProbabilityDistributions
Section 1: Probability
Contents: 1.1 1.2 1.3 1.4 1.5 Some basics of probability theory Axioms, Interpretations, and Properties of Probability Counting Techniques and Probability Conditional Probability Independence
TEM1116
1
TEM1116 Probability and Statistics
Tri1 2013/14
Chapter 1
1.1
Basics of Probability Theory
Probability refers to the study of randomness and uncertainty. The word “probability” as used in “probability of an event” is a numerical measure of the chance for the occurrence of an event. Experiment: a repeatable procedure with a welldefined set of possible outcomes. (Devore: Any action or process whose outcome is subject to uncertainty.) Sample Space and Events Sample space of an experiment is the set of all possible outcomes. An event is a set of outcomes (it is a subset of the sample space). Example: Consider an experiment of rolling a 6sided die.
Sample Space, S :
{1, 2, 3, 4, 5, 6}
S
Events, Ek: E1: composite number is rolled. → Equivalently, {4, 6}. E2: number less than four is rolled. → Equivalently, {1, 2, 3}.
E1
E2
Example 1.1 : An experiment consists of tossing three...
...Some Important DiscreteProbabilityDistributions
51
Chapter Goals
After completing this chapter, you should be able
to:
Interpret the mean and standard deviation for a
discreteprobabilitydistribution
Explain covariance and its application in finance
Use the binomial probabilitydistribution to find
probabilities
Describe when to apply the binomialdistribution
Use Poisson discreteprobabilitydistributions to
find probabilities
52
Definitions
Random Variables
A random variable represents a possible
numerical value from an uncertain event.
Discrete random variables produce outcomes
that come from a counting process (e.g. number
of courses you are taking this semester).
Continuous random variables produce outcomes
that come from a measurement (e.g. your annual
salary, or your weight).
53
Definitions
Random Variables
Random
Variables
Ch. 5
Discrete
Random Variable
Continuous
Random Variable
Ch. 6
54
Discrete Random Variables
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 could be 0, 1, or 2 times)
Toss a coin 5 times.
Let X be the number of heads
(then X = 0, 1, 2, 3, 4, or 5)
55...
...Probabilitydistribution
Definition with example:
The total set of all the probabilities of a random variable 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 random variable. Now what are the possible values of the random variable, 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 random variable 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 random variable had 3 values namely 0, 1, and 2. These are discrete values. It might happen in 1 certain example that 1 random variable assumes 1 continuous range of values between x to y. In that case also we can find the probabilitydistribution of the random variable. Soon we shall see that there are three types of probabilitydistributions. Two of them deal...
...EXERCISES (DiscreteProbabilityDistribution)
EXERCISES (DiscreteProbabilityDistribution)
P X x n C x p 1 p
x
BINOMIAL DISTRIBUTION
n x
P X x n C x p 1 p
x
BINOMIAL DISTRIBUTION
n x
1. 2. 3.
The probability that a certain kind of component will survive a given shock test is ¾. Find theprobability that exactly 2 of the next 4 components tested survive. The probability that a logon to the network is successful is 0.87. Ten users attempt to log on, independently. Find the probability that between 4 and 8 logons are successful. The probability that a patient recovers from a rare blood disease is 0.40. If 15 people are known to have contracted this disease, what is the probability that (a) at least 10 survive, (b) from 3 to 8 survive, and (c) exactly 5 survive? NEGATIVE BINOMIAL DISTRIBUTION
1. 2. 3.
P X x k 1 C x1 p 1 p
x
The probability that a certain kind of component will survive a given shock test is ¾. Find the probability that exactly 2 of the next 4 components tested survive. The probability that a logon to the network is successful is 0.87. Ten users attempt to log on, independently. Find the probability...
...The Poisson probabilitydistribution, named after the French mathematician SiméonDenis. Poisson is another important probabilitydistribution of a discrete random variable that has a large number of applications. Suppose a washing machine in a Laundromat breaks down an average of three times a month. We may want to find the probability of exactly two breakdowns during the next month. This is an example of a Poisson probabilitydistribution problem. Each breakdown is called an occurrence in Poisson probabilitydistribution terminology. The Poisson probabilitydistribution is applied to experiments with random and independent occurrences. The occurrences are random in the sense that they do not follow any pattern, and, hence, they are unpredictable. Independence of occurrences means that one occurrence (or nonoccurrence) of an event does not influence the successive occurrences or nonoccurrences of that event. The occurrences are always considered with respect to an interval. In the example of the washing machine, the interval is one month. The interval may be a time interval, a space interval, or a volume interval. The actual number of occurrences within an interval is random and independent. If the average number of occurrences for a given interval is known, then by using the Poisson...
...ProbabilityDistribution Essay
Example Suppose you flip a coin two times. This simple statistical experiment can have four possible outcomes: HH, HT, TH, and TT. Now, let the random variable X represent the number of Heads that result from this experiment. The random variable X can only take on the values 0, 1, or 2, so it is a discrete random variable
Binomial Probability Function: it is a discretedistribution. The distribution is done when the results are not ranged along a wide range, but are actually binomial such as yes/no. This is used frequently in quality control, reliability, survey sampling, and other corporate and industrial situations. This type of distribution can measure levels of performance only if the results can be placed into a binomial order, such as with a point estimate where only one number is relied upon. For example, if you measure whether unit X had exceeded its monthly energy limits usage and is interested in a yes or no answer. This type of distribution gives the probability of an exact number of successes in independent trials (n), when the probability of success (p) on single trial is a constant.
The probability of getting exactly r success in n trials, with the probability of success on a single trial being p is:
P(r) (r successes in n trials) = nCr . pr ....