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STA247 (Conditional Probability (marginal probability (so when we take…

- - - - I double-checked my argument and her homework assignment question, and I am right, the bottom only requires y, so when y is given as a condition, it doesn't matter if we apply it before the integral or take it out.
        
        Intuitively, this also makes sense because the conditional probability is simply how much proportion that line interval takes up along that whole line of support in that given y-value position.
- - - - It's like, a distribution has a certain average, and certain variance among every trial. But when we do sampling, what we observe might be different. So in sampling distribution, we often 'recognize' the distribution first, then we predict what sampling might look like for each sampling, with sample size n=... and when we do the simulations a lot of times, we expect to get an average of it with the sampling mean(predicted). But the question is how do we truly recognize any distribution in the first place other than the Bernoulli trial? Even tossing coins is not 50 50 in real life because of some small dents and different angles.
- - - - made possible through natural log, e^x's derivative is simply an x in front. Also we, but Leibniz's property shows that taking derivative of the integral allows us to obtain E[X^2], E[X^3] and so on, because Leibiniz's rule allows the derivative to penetrate the integral sign to obtain the resulting moments. So using Leibniz, we should that taking derivatives of the function is actually the same as calculating the moments.
      - we can use it on Binomial Distributed RV too, in fact, check the list of Distributions where it can be applied
    - - we can use property two as way of identifying distribution for function of Random Variable. Y= X1 + X2 +...+Xn
- - - - Exponential Distribution
        
        Gamma Distribution
        
        The gamma distribution is often used to measure waiting times such as time until “death/failure”. It is more flexible than the exponential distribution (a = 1), and unlike the exponential, the gamma random variable is not memoryless.
        
        Poisson and Exponential models the same distribution but focuses on different aspects of the problem
        
        memorylessness
      - The Poisson distribution is appropriate since copper flaws are representative of events that occur over a continuous interval of measure. The probability that a flaw occurs at any point on the wire is consistent and each section of wire is independent.
- - - - (?->) Chebyshev's Inequality
        
        Normal Distribution(also within # of standard deviations')