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Probability (3: Conditional probabilities (Baye's Formula: IMG…

- - - - \(X_1 \sim Pois(\lambda_1), X_2 \sim Pois(\lambda_2); then X_1+X_2 \sim Pois(\lambda_1+\lambda_2)\)
      - \(X_1 \sim Bin(n,p), X_2 \sim Bin(m,p); then X_1+X_2 \sim Bin(n+m,p)\)
- - - - \(E[X] = \frac{\beta +\alpha}{2}, Var(X) = \frac{(\beta -\alpha)^2}{12}\)
    - - \(E[X]=\mu, Var(X)=\sigma ^2\) The Normal Approximation to the Binomial Distribution
    - - \(E[X]= \frac{1}{\lambda}; Var(X)= (\frac{1}{\lambda})^2\) In practice, the exponential distri often arises as the distri of the amt of time until some specific event occurs.
  - - - E[X] = np; Var(X) = np(1-p)
      - Two approximations: :check: Poisson when n large p small & np moderate; :check: Nomal when np(1-p) large (\(\geq 10\))
      - Proposition 6.1 If X is a binomial rv with parameters (n,p), where 0<p<1, then as k goes from 0 to n, P(X=k) first increases monotonically and then decreases monotonically, reaching its largest value when k is the largest integer \(\leq\) (n+1)p
    - - Tremendous range of applications in diverse areas because it may be used as approximation for a binomial rv~(n,p) when n is large enough and p is small enough so that np is of moderate size. \(\lambda\) = np = E(# of successes), apply both when n independent trails and trials have a weak independence
      - E[X] = \(\lambda\); Var(X)= \(\lambda\)
      - Poisson Paradigm: Consider n events, with \(p_i\) = prob that event i occurs, i = 1,...,n. If all the \(p_i\) are "small" and the trials are either independent or at most "weakly dependent," then the # of these events that occur approximately has a Poisson distribution with mean \(\sum_{i=1}^np_i\)
      - Another use of Poisson arises in situations where "events" occur at certain pts in time.
      - We say that the events occur in accordance with a Poisson process having rate \(\lambda\). The value \(\lambda\), which can be shown = the rate per unit time at which events occur, is a constant that must be empirically determined.
    - - \(E[X] = \frac{1}{p}, Var(X) = \frac{1-p}{p^2}\)
    - - \(E[X] = \frac{r}{p}; Var(X) = \frac{r(1-p)}{p^2}\)
      - \(E[X]=\frac{nm}{N}; Var(X)=np(1-p)(1-\frac{n-1}{N-1})\)
    - - For a rv X, let X(s) denote the value of X when \(s\in S\) is the outcome of the experiment. If X and Y are both rv, so it their sum. Proposition 9.1: \(E[x]=\sum_{s\in S}X(s)p(s)\)
      - Corollary 9.2: The expected value of a sum of rv = the sum of their expectations.
    - - Step fcn where the size of the step of any of the values = the prob that X assumes that particular value.
    - - Proposition 4.1. If X is a discrete rv. that takes on one of the values \(x_i, i\geq1\), with respective prob \(p(x_i)\), then, for any real-valued fcn g, \(E[g(X)] = \sum_i g(x_i)p(x_i)\)
      - Corollary 4.1 If a and b are constants, then E[aX+b] = aE[X] +b
      - The nth moment of X \(E[X^n] = \sum_{x:p(x)>0}x^np(x)\) E[X] is also referred as the mean/ the 1st moment of X