The weekly repair cost, X, for a certain machine has a probability density function given by 6x(1 – x), 0

The probability density function of X, the lifetime of a certain type of device (measured in months), is given by so if x 5 x2 Find the following: P(X > 9) = The cumulative distribution function of X: if x 5 The probability that at least one out of 5 devices of this type will function for at least 26 months:

The Gamma distribution is often used to describe the wait time until a certain number of events occurs. Assume x1, . . . , xn ∼iid Gamma(α, β), where α is shape and β is rate. Calculate the maximum likelihood estimator of β either by hand or via R, but not both. The probability density function (PDF) for the Gamma distribution is: Calculate the log-likelihood function of the function below. I t doesnt have to be on r studi you can do by your hand

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.