Download Discrete Probability Distributions

Discrete Probability
Distributions
Random Variable
 Random variable is a variable whose value is subject to
variations due to chance. A random variable conceptually does
not have a single, fixed value (even if unknown); rather, it can
take on a set of possible different values, each with an
associated probability.
Discrete Random Variable
Continuous Random Variable
Discrete Random Variables
Discrete Probability Distribution
Discrete Probability Distribution
Discrete Random Variable Summary
Measures
 Expected Value : the expected value of a random variable is
the weighted average of all possible values that this random
variable can take on. The weights used in computing this average
correspond to the probabilities in case of a discrete random
variable, or densities in case of a continuous random variable ;
Discrete Random Variable Summary
Measures
 Standard deviation shows how much variation or "dispersion"
exists from the average (mean, or expected value);
Discrete Random Variable Summary
Measures
Probability Distributions
The Bernoulli Distribution
 Bernoulli distribution, is a discrete probability distribution,
which takes value 1 with success probability p and value 0
with failure probability q=1-p .
 The Probability Function of this distribution is;
The Bernoulli distribution is simply Binomial (1,p)
.
Bernoulli Distribution Characteristics
The Binomial Distribution
Counting Rule for Combinations
Binomial Distribution Formula
Binomial Distribution
Binomial Distribution Characteristics
Binomial Characteristics
Binomial Distribution Example
Geometric Distribution
 The geometric distribution is either of two discrete
probability distributions:


The probability distribution of the number of X Bernoulli
trials needed to get one success, supported on the
set { 1, 2, 3, ...}
The probability distribution of the number Y = X − 1 of
failures before the first success, supported on the
set { 0, 1, 2, 3, ... }
Geometric Distribution
 It’s the probability that the first occurrence of success require k
number of independent trials, each with success probability p. If
the probability of success on each trial is p, then the probability
that the kth trial (out of k trials) is the first success is
 The above form of geometric distribution is used for modeling
the number of trials until the first success. By contrast, the
following form of geometric distribution is used for modeling
number of failures until the first success:
Geometric Distribution Characteristics
The Poisson Distribution
Poisson Distribution Formula
Poisson Distribution Characteristics
Graph of Poisson Probabilities
Poisson Distribution Shape
The Hypergeometric Distribution
Hypergeometric Distribution Formula
Hypergeometric Distribution Example
Continuous Probability
Distributions
Continuous Probability Distributions
The Normal Distribution
Many Normal Distributions
The Normal Distribution Shape
Finding Normal Probabilities
Probability as Area Under the Curve
Empirical Rules
The Empirical Rule
Importance of the Rule
The Standart Normal Distribution
The Standart Normal
Translation to the Standart Normal
Distribution
Example
Comparing x and z units
The Standart Normal Table
The Standart Normal Table
General Procedure for Finding
Probabilities
z Table Example
z Table Example
Solution : Finding P(0 < z <0.12)
Finding Normal Probabilities
Finding Normal Probabilities
Upper Tail Probabilities
Upper Tail Probabilities
Lower Tail Probabilities
Lower Tail Probabilities
The Uniform Distribution
The Uniform Distribution
The Mean and the Standart Deviation
for Uniform Distribution
The Uniform Distribution
The Uniform Distribution
 Characteristics;
The Exponential Distribution
The Exponential Distribution
Shape of the Exponential Distribution
Example