Gamma Distribution,Poisson Process,Exponential Distribution & Memoryless Property Explained Clearly скачать в хорошем качестве

Gamma Distribution,Poisson Process,Exponential Distribution & Memoryless Property Explained Clearly

Gamma Distribution,Poisson Process,Exponential Distribution & Memoryless Property Explained Clearly Linkedin   / pratap-padhi   Website https://smearseducation.com/ Join my FREE Skool Community to get all updates and support https://www.skool.com/sme-education-9... Watch my previous recordinds on CS2 Time Series 👉    • Master Time Series Forecasting:Guide to AR...   CS2 Risk Modelling and Survival Analysis 👉    • What is a Stochastic Process? Easy explana...   CS1 Previous recorded videos watch 👉    • What are discrete random variables? |Class...   CM1 Previous recorded videos watch 👉    • How to calculate simple interest | Fundame...   TIMESTAMPS 00:00 Class introduction and learning objectives 00:30 Why gamma distribution matters across CS1, CS2, CM2 01:02 Shape and scale parameters. Graph intuition 01:27 How gamma changes shape toward normal 01:51 Why factorial fails and gamma function is needed 02:28 Gamma function definition and intuition 03:05 Evaluating gamma for integer values 04:47 Recursive property of gamma function 05:52 Why gamma works for non-integers 06:48 Why gamma(1/2) is special 08:02 Practical takeaway from gamma function 11:14 Moving from gamma function to gamma distribution 12:09 Gamma density and positive skewness 13:03 Real-life meaning of positive skew 13:52 Why total probability equals one 15:41 Mean and variance of gamma distribution 19:46 General trick for computing expectations 23:29 Deriving EX and EX² clearly 27:46 Variance formula of gamma 28:05 Memoryless property intuition 29:18 Exponential as a special case of gamma 30:18 Bulb lifetime example 31:47 Formal memoryless property proof idea 34:31 Why gamma generalizes exponential 35:26 Poisson distribution vs Poisson process 36:30 Time dependency in Poisson process 37:28 Counting process intuition 38:19 Inter-arrival times explained 39:44 Why waiting times are exponential 40:25 From exponential waiting times to gamma 41:13 Real-world bus arrival example 42:09 Discrete vs continuous connection 43:10 Summary and next topics DETAILED VIDEO DESCRIPTION This class builds one of the most important conceptual bridges in probability and stochastic processes. Gamma distribution, Poisson process, exponential distribution, and the memoryless property are not separate topics. They are deeply connected. The session starts with gamma distribution from a graphical and intuitive perspective. You learn how the shape and scale parameters control skewness and spread. You see how gamma slowly transitions toward a normal-like shape as the shape parameter increases. Before touching gamma distribution, the class explains gamma function. Why factorial fails for non-integers. Why probability theory needs an extension. How gamma function generalizes factorial for continuous values. Why values like 1.5 factorial exist and why gamma(1/2) equals √π. With it, gamma distribution feels inevitable. You then move into gamma distribution itself. You learn why it is positively skewed. What positive skew means in real life. Why high values have low probability but still matter. How total probability integrates to one and why gamma function appears naturally in the density. The class then derives the mean and variance of gamma distribution step by step. Not as memorisation, but by converting the density into another gamma density. This technique becomes extremely useful later in CS2 and CM2. Next comes the conceptual heart of the lecture. The memoryless property. You learn why exponential distribution is special. Why past usage does not affect future lifetime. Why bulb lifetimes and waiting times behave differently from aging models. How the memoryless property appears mathematically and intuitively. Then the class connects exponential distribution to gamma distribution. Exponential is not separate. It is a special case of gamma. Finally, the class introduces Poisson process. Not just Poisson distribution, but time-evolving counting processes. You learn • Difference between Poisson distribution and Poisson process • What a rate really means • Why arrivals are independent • Why inter-arrival times follow exponential distribution • Why waiting for multiple events leads to gamma distribution Bus arrivals, claim arrivals, customer arrivals, and student join times are used as examples. By the end of this class, you clearly understand • Why Poisson counts events • Why exponential measures waiting time • Why gamma measures waiting until the k-th event • How all three are mathematically and conceptually linked #Probability #PoissonProcess #GammaDistribution #ExponentialDistribution #WaitingTime #ActuarialScience #Statistics #StochasticProcesses #DataScience #OperationsResearch