5. When a muon decays, an electron is emitted at an angle 0 relative to the spin, such that the probability density function of the angular distribution is dP = k(1+ax) dx, where x = cos 0 and a is a constant that must be determined experimentally. Find the value of the constant k, and the mean and variance of the x distribution (in terms of a).

The Exponential pdf for continuous random variable Y takes the form e-ylß y > 0 f(V) = for ß > 0. y 1.6) Give your answer to three decimal places.

2. In probability, it is common to model the deviation of a day's temperature from the monthly average temperature using the Gaussian probability density function, 1.-/9 f(t) = V9T This means that the probability that the day's temperature will be between t = a and t = b different from the monthly average temperature is given by the area under the graph of y = f(t) between t a andt = b. A related function is %3D F(x) = -2/9 dt, 2 0. %3D V9n Jo This function gives the probability that the day's temperature is between t = -x and t = x different from the monthly average temperature. For example, F(1) 0.36 indicates that there's roughly a 36% chance that the day's temperature will be within 1 degree (between 1 degree less and 1 degree more) of the monthly average. 1 (a) Find a power series representation of F(x) (write down the power series using sigma notation). (b) Use your answer to (a) to find a series equal to the probability that the day's temperature will be within 2 degrees of the…

Suppose that the interval between eruptions of a particular geyser can be modelled by an exponential distribution with an unknown parameter 0 > 0. The probability density function of this distribution is given by f(x; 0) = 0e 0¹, x > 0. The four most recent intervals between eruptions (in minutes) are x₁ = 32, x₂ = 10, x3 = 28, x4 = 60; their values are to be treated as a random sample from the exponential distribution. (a) Show that the likelihood of based on these data is given by L(0) 04-1306 = (b) Show that L'(0) is of the form L'(0) = 0³ e 1300 (4- 1300). (c) Show that the maximum likelihood estimate of 0 based on the data is ~ 0.0308 making your argument clear. (d) Explain in detail how the maximum likelihood estimate of that you have just obtained in part (c) relates to the maximum likelihood estimator of for an exponential distribution.