B1 The x distribution is a special case of Gamma distribution (not to be confused with gamma function; see below). The density function of the Gamma distribution with parameters and k is given by where -1 e -x/0 (x) = = г(k) Øk if x > 0, and otherwise, г(k) = √ ₁ k-1-x dx x' e is the gamma function. (a) For every k ≥ 1, 0 > 0, find the mode of the density. Hint: The algebra can be simplified by appropriate use of logarithms. ~ Now suppose that X1,..., Xn id Exp(\) and that we have a prior belief in A which is consistent with a prior distribution X. Gamma(a, b), for some a, ß, i.e. the prior density of is Baxa-1-BA T(a) e = so 01/ẞ and k = a. (b) Write down the likelihood, and show that the posterior distribution for \ is also a Gamma distribution, but with parameters a +n and B + Σ Xi. (c) Find the mode of the posterior distribution and examine the behaviour as n → ∞.

The probability density function of a distribution is given by x f(x) = exp(-7). Use differentiation to show that the probability density function has a maximum at x = 0. The moment generating function of a distribution is M(t) = (q + pet)", where p € [0, 1], q = 1 - p and n is a positive integer. Use the moment generating function to find the mean and variance of the distribution in terms of p, q and n.

inclue the variance

Suppose that a measures the time (in hours) it takes for a student to complete a comprehensive exam. All students are done within 3 hours and the density function for x is if 0

I would like a copy of the answer typed on the keyboard, please

2. Identify the probability density function, then find the mean and variance without integrating. b. f(x) =1/6 e^−x/6,   [0,∞) c. f(x) =1 / 3√2π e^−(x−16)^2/18,  (−∞,∞)

Suppose that the graph below represents the density function for the times (in weeks) that Canadians will need to wait before receiving a COVID-19 vaccine, starting today, as predicted by the Canadian goverment. y = p(t) Wait Time t (weeks) 3 (a) Given that p is a density function, what numerical value should be placed in the pink box? The value in the pink box is mumber units (b) For each question below, please write your answer into the box. You do not need to explain your reasoning. i. What fraction of Canadians will receive their vaccine within 2 weeks from today? (Please state your answer in the form of a fraction.). ii. What is the median wait time? iii. Without doing any calculations, how do you expect the mean wait time to compare with the median? (Please fill in the blank below using the words "greater than", "less than", or "equal to".) weeks I expect the mean wait time to be the median. iv. According to this model, what is the longest possible time that a Canadian might need…

Please don't provide handwrittin solution...

Let X a random variable with density function = e- . 1(0,00) (x). Determine tha density function of Y = 1+X

pls include the variance