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, ... }
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
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