for x \ge 0 x ≥0, \alpha > 0 α > 0 and \sigma > 0 σ > 0 . (Here \Gamma (\alpha) Γ(α) is the function implemented by R 's gamma () and defined in its help. Note that a = 0 a = 0 corresponds to the trivial distribution with all mass at point 0.) The mean and variance are E (X) = \alpha\sigma E (X) =ασ and Var (X) = \alpha\sigma^2 Var(X
We can get the probability that 5 people in a random 10-person sample would smoke using: SAS: R: PDF(’Binomial’, 5, .22, 10); dbinom(5, size = 10, prob = .22) This returns 3.7%, the same result we obtained in class. In R, you can leave out the size = and prob = , but if you do so, you have to have everything in a particular order. For example: Binomial Theorem. Example 25.1.4. Recall the well-known binomial formula: (a + b)2 = a2 + 2ab + b2. (since, using ``FOIL,'' we have: (a + b)2 = (a + b) ⋅ (a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2) In this section we generalize this to find similar expressions for (a + b)n for any natural number n. This is the content of the (generalized Definition of P-value is the probability of getting “at least as extreme dataset under the null hypothesis as was observed”. The logic is that if P-value is very small, then it would be very improbable to observe a data set that is at least as extreme as was observed if the null hypothesis was true. Let’s see how to plot binomial distribution in R. First need to create a probability mass function while using dbinom(x, size, prob) plot(x, y, type = ‘h’) to plot the probability mass function. As discussed earlier we need to mention the number of trials and probability of success on a given trial in the dbinom() function. oNhwE.