X of a certain chemical reaction, measured in hours, has the probability density function f(x) = -2x + 2 over the interval [0,1]. A. Draw the draw of this probability density function.  B. Verify that it is a probability density function of a continous random variable X.  C. Find the probability that the time

For the probability density function f defined on the random variable x, find (a) the mean of x, (b) the standard deviation of x, and (c) the probability that the random variable x is within one standard deviation of the mean. f(x) = 195x, (0,5] 2, [0,5] 125 a) Find the mean. (Round to three decimal places as needed.) b) Find the standard deviation. O = (Round to three decimal places as needed.) c) Find the probability that the random variable x is within one standard deviation of the mean. The probability is. (Round to three decimal places as needed.)

(c) At a bus stop, there are two buses available (bus numbers 1 and 2). For i = 1,2, the waiting time until bus i arrives is denoted by T;, with T independent of T2, and T; has density function fr. (t) = X, exp(-1;t), t>0.

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