Tutorial on Discrete Probability Distributions
Tutorial on discrete probability distributions 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. A discrete probability distribution 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 probability distribution 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 probability distribution of X is given by X P(X) 0 1 / 4
1 1 / 2
2 1 / 4
* * Note that ∑ P(X) = 1 * b) The mean µ of the random variable X is defined by µ = ∑ X P(X)
= 0 * (1/4) + 1 * (1/2) + 2 * (1/4) = 1 * The standard deviation σ of the random variable X is defined by σ = Square Root [ ∑ (X µ) 2 P(X) ]
= 1 / square root (2)Example 2:Two balanced dice are rolled. Let X be the sum of the two dice.
a) Obtain the probability distribution of X.
b) Find the mean and standard deviation of X.Solution to Example 2: * a) When the two balanced dice are rolled, there are 36 equally likely possible outcomes as shown below . * The possible values of X are: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. * The possible outcomes are equally...
...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...
...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
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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)
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Probability...
...Discrete and Continuous Probability
All probabilitydistributions can be categorized as discreteprobabilitydistributions or as continuous probabilitydistributions (stattrek.com). A random variable is represented by “x” and it is the result of the discrete or continuous probability. A discreteprobability 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...
...TEM1116 Probability and Statistics
Tri1 2013/14
Chapter 1
Chapter 1: Discrete and Continuous ProbabilityDistributions
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 coins. Find the...
...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 with...
...If a six sided die is tossed two times and “3” shows up both times, the probability of “3” on the third trial is
a. much larger than any other outcome
b. much smaller than any other outcome
c. 1/6
d. 1/216
13. If P(A) = 0.4, P(B A) = 0.35, P(A B) =0.69, then P(B) =
a. 0.14
b. 0.43
c. 0.75
d. 0.59
14. Two events with nonzero probabilities
a. can be both mutually exclusive and independent
b. can not be both mutually exclusive and independent
c. are always mutually exclusive
d. are always independent
15. If A and B are independent events with P(A) = 0.05 and P(B) = 0.65, then
P(AB) =
a. 0.05
b. 0.0325
c. 0.65
d. 0.8
16. A description of the distribution of the values of a random variable and their associated probabilities is called a
a. probabilitydistribution
b. random variance
c. random variable
d. expected value
The following represents the probabilitydistribution for the daily demand of microcomputers at a local store.
Demand Probability
0 0.1
1 0.2
2 0.3
3 0.2
4 0.2
17. Refer to above information, the expected daily demand is
a. 1.0
b. 2.2
c. 2, since it has the highest probability
d. of course 4, since it is the largest demand level
18. Refer to above information, the probability of having a demand for at least two...
...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...
...CHAPTER 3: PROBABILITYDISTRIBUTION
3.1
RANDOM VARIABLES AND PROBABILITYDISTRIBUTION
Random variables is a quantity resulting from an experiment that, by chance, can assume different values. Examples of random variables are the number of defective light bulbs produced during the week and the heights of the students is a class. Two types of random variables are discrete random variables and continuous random variable.
3.2
DISCRETE RANDOM VARIABLE
A random variable is called a discrete random variable if its set of posibble outcomes is countable. Probabilitydistribution is a listing of all the outcomes of an experiment and the probability associated with each outcome. For example, the probabilitydistribution of rolling a die once is as below: Outcome, x Probability, P(x) 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6
The probabilitydistribution for P(x) for a discrete random variable must satisfy two properties: 1. The values for the probabilities must be from 0 to 1; 0 ≤ ( ) ≤ 1 2. The sum for P(x) must be equal to 1; ∑ ( ) = 1
QMT200
3.2.1 FINDING MEAN AND VARIANCE Mean of X is also referred to as its “expected value”.
= ( ) Where: = ∑[ ( )]
( )=
= (
) − [ ( )]
(
)=
[
( )] = ( )
Example 1 An...