# Chapter 7 : Discrete Probability

### Topics covered in this snack-sized chapter:

#### Discrete Probability arrow_upward

• The Probability Distribution for a random variable describes how probabilities are distributed over the values of the random variable.
• We can describe a Discrete Probability Distribution with a table, graph, or equation.
• Discrete outcomes are:
• Mutually exclusive (nothing in common).
• Collectively exhaustive (nothing left out).
• Discrete Probability ranges from 0 to 1.
•  • Discrete Probability Distribution includes the following:
• #### Random Variables arrow_upward

• A Random Variable is a numerical description of the outcome of an experiment.

##### Discrete Random Variable:
• A Discrete Random Variable may assume either a finite number of values or an infinite sequence of values.
• ##### Continuous Random Variable:
• A Continuous Random Variable may assume any numerical value in an interval or collection of intervals.

• #### Expected Value (The Mean) arrow_upward

• It is a weighted average of the Probability Distribution.
• Multiplying the P of each outcome by the value of the outcome, and then summing the results.
•  ##### Example:
• Toss 2 coins, count the number of tails and compute expected value.
• Solution:

•  Values Probability 0 0.25 1 0.50 2 0.25  #### Variance arrow_upward

• The weighted average squared deviation about the mean.
•  ##### Example:
• Toss 2 coins, count the number of tails and compute variance.
• ##### Solution:

 Values Probability 0 0.25 1 0.50 2 0.25 #### Standard Deviation arrow_upward

• The Standard Deviation is the square root of the Variance.
• #### Co-variance arrow_upward

• Co-variance is a statistic representation of the degree to which two variables vary together.
• It shows the type and strength of the relationship between two variables.
• Positive
• Indicates that the two variables will vary together in the same direction.
• Negative
• Indicates that the two variables will vary in opposite directions. Where, Discrete Random Variable. Outcome of . Discrete Random Variable. Outcome of . Probability of occurrence of the outcome of and the outcome of .

#### Binomial Probability Distribution arrow_upward

• It is used to find the probability that a given number of successes X occurs over a certain number of trials n, each of which has the same probability p.
• Success refers to the event of interest to us, such as:
• Getting a head when flipping a coin.
• Getting a question correct on a quiz.

#### Binomial Distribution: Formula arrow_upward

• Let x denote the number of successes occurring in the n trials.
• • Where,
• f(x) = The probability of x successes in n trials.
• p = The probability of success on any one trial.

#### Mean of Binomial Distribution arrow_upward

• Let X be a discrete random variable with the binomial distribution with parameters n and p.
• The mean of binomial Distribution is given by:
• #### Variance and Standard Deviation of Binomial Distribution arrow_upward

• Let X be a discrete random variable with the binomial distribution with parameters n and p.
• The variance and standard deviation of binomial Distribution is given by:
• • • #### Poisson Distribution arrow_upward

• The Poisson Distribution is used to model the number of events occurring within a given time interval.

• ###### Properties of Poisson Experiment:

• The probability of an occurrence is the same for any two intervals of equal length.
• The expected value of occurrences in an interval is proportional to the length of this interval.
• The occurrence or non-occurrence in any interval is independent of the occurrence or non-occurrence in any other interval.
• The probability of two or more occurrences in a very small interval is close to 0.

• #### Poisson Distribution: Formula arrow_upward Where, is an average rate of value.
x is a poisson random variable.
e is the base of logarithm (e = 2.718).

#### Hyper-Geometric Distribution arrow_upward

• The Hyper-Geometric Distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement.
• • Where,
• N = Population size.
• A = Number of successes in the population.
• N – A = Number of failures in the population.
• n = Sample size.
• X = Number of successes in the sample.
• n – X = Number of failures in the sample.

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