Probability Distribution Types PDF

© DatabaseTown.com PROBABILITY DISTRIBUTION A distribution represent the possible values a random variable can take and how often they occur. Mean – it represent the average value which is denoted by µ (Meu) and measured in seconds Variance – it represent how spread o...

© DatabaseTown.com PROBABILITY DISTRIBUTION A distribution represent the possible values a random variable can take and how often they occur. Mean – it represent the average value which is denoted by µ (Meu) and measured in seconds Variance – it represent how spread out the data is, denoted by 𝜎 2 (Sigma Square). It is pertinent to note that it cannot be measured in seconds square which make no sense,therefore, variance is measured by Standard Deviation which is the square root of variance √σ2andhas the same unit as means. There are two kinds of data i.e. population data and sample data. Population Sample Data Vs Data “all the data” “just a part of it” Population Data Sample Data Mean µ 𝑥̅ Variance 𝜎2 S2 Standard Deviation 𝜎 S https://databasetown.com Page 1  © DatabaseTown.com The more overfilled the mid of the distribution, the more data falls within that interval as show in figure P µ-𝜎 µ+𝜎 y The fewer data falls within the interval, the more spread the data is, as shown in figure P µ-𝜎 µ+𝜎 y Notation of Distributions: Y – Actual outcome y – one of the possible outcomes P(Y=y) – Probability distribution which is equal to p(y) TYPES OF DISTRIBUTIONS: Two major kind of distributions based on the type of likely values for the variables are, A. Discrete Distributions B. Continuous Distributions https://databasetown.com Page 2  © DatabaseTown.com COMPARISON BETWEEN DISCRETE AND CONTINUOUS DISTRIBUTIONS: Discrete Distributions Continuous Distribution Discrete distributions have finite number of Continuous distributions have infinite many different possible outcomes consecutive possible values We can add up individual values to find out the We cannot add up individual values to find out probability of an interval the probability of an interval because there are many of them Discrete distributions can be expressed with a Continuous distributions can be expressed with graph, piece-wise function or table a continuous function or graph In discrete distributions, graph consists of bars In continuous distributions, graph consists of a lined up one after the other smooth curve Expected values might be unachievable To calculate the chance of an interval, we required integrals Notation Explanation: X ~ 𝑁 (µ, 𝜎2) Here, X is variable, ~ tilde, N is types of distribution and (µ, 𝜎2) are its characteristics A. DISCRETE DISTRIBUTIONS: Discrete distributions have finite number of different possible outcomes. Its main characteristics are given below:-  We can add up individual values to find out the probability of an interval  Discrete distributions can be expressed with a graph, piece-wise function or table  In discrete distributions, graph consists of bars lined up one after the other  Expected values might be unachievable  P(Y≤y) = P(Y < y + 1) In graph, the discrete distributions are looks like as, https://databasetown.com Page 3  © DatabaseTown.com Examples of Discrete Distributions: i. Bernoulli Distribution ii. Binomial Distribution iii. Uniform Distribution iv. Poisson Distribution i. Bernoulli Distribution: In Bernoulli distribution there is only one trial and only two possible outcomes i.e. success or failure. It is denoted by y ~Bern(p). The main characteristics of Bernoulli distributions are:  It consists of a single trial  Two possible outcomes  E(Y) = p  Var(Y) = p × (1 – p) Examples and Uses:  Guessing a single True/False question  It is mostly used when trying to find out what we expect to obtain a single trial of an experiment. ii. Binomial Distribution: A sequence of identical Bernoulli events is called Binomial and follows a Binomial distribution. It is denoted by Y ~ B(n, p). The main characteristics of Binomial distribution are:  Over the n trials, it measures the frequency of occurrence of one of the possible result.  E(Y) = n × p  P(Y = y) = C(y, n) × py× (1 – p)n-y  Var(Y) = n × p × (1 – p) Examples and Uses:  Simply determine, how many times we obtain a head if we flip a coin 10 times. https://databasetown.com Page 4  © DatabaseTown.com  It is mostly used when we try to predict how likelihood an event occur over a series of trials. iii. Uniform Distribution: In uniform distribution all the outcomes are equally likely. It is denoted by Y ~ U(a, b). If the values are categorical, we simply indicate the number of categories, like Y ~ U(a). The main characteristics of Uniform Distribution are:  In uniform distribution all the outcomes are equally likely.  In graph, all the bars are equally tall  The expected value and variance have no predictive power Examples and Uses:  Result obtained after rolling a die  Due to its equality, it is mostly used in shuffling algorithms iv. Poisson Distribution: Poisson distribution is used to determine how likelihood a certain event occur over a given interval of time or distance. It is denoted by Y ~ Po(λ). The main characteristics of poisson distribution are:  It measures the frequency over an interval of time or distance.  E(Y) = 𝜆 𝜆𝑦  P(Y = y) = 𝜆!𝑒 −𝜆 https://databasetown.com Page 5  © DatabaseTown.com  Var(Y) = 𝜆 Examples and Uses:  It is used to determine how likelihood a certain event occur over a given interval of time or distance.  Mostly used in marketing analysis to find out whether more than average visits are out of the ordinary or otherwise. B. CONTINUOUS DISTRIBUTIONS: Continuous distributions have infinite many consecutive possible values. Its main characteristics are given below:-  We cannot add up individual values to find out the probability of an interval because there are many of them  Continuous distributions can be expressed with a continuous function or graph  In continuous distributions, graph consists of a smooth curve  To calculate the chance of an interval, we required integrals  P(Y = y) = 0 for any distinct value y.  P(Y 1 then E(Y) = µ and Var(Y) = s2 × 𝑘−2 Examples and Uses:  It is used in examination of a small sample data which normally follows a normal distribution. https://databasetown.com Page 10