Let U1, . . . , Un be a random sample from the Unif(0,1) distribution. (a) Find the distribution of Y1 = −2 log U1 .(b) Show that the distribution Y = −2 ∑ni=1 log Ui is χv2 for some value of ν. (c) What is the value of ν in part (b)?

12) A metal bar is heated, and then allowed to cool. Its temperature T (in °C) is found to be The time is in minutes. T=15+75e-0.25 t Find the time rate of change of temperature after 5.0 minutes. 13) The standard normal curve (sometimes called the bell curve) in statistics looks like: 1 -x² y = e 2 √2π Show that there are two inflection points at x = +1

The random variable is said to follow the distribution ofBenford’s Law if P(X=i)=log10(1+i)/i find P(X<= j) ?

iid *1. Let Y1,..., Yn * uniform(0, 1). Find the distribution of their geometric mean: U = i=1 (Hint: consider first finding the distribution of –n log U.) - N (µ, o2). Find the distribution of Y and show that P(Y s e") = 0.5 (i.e., e" is *2. Let log Y the median of Y).

Let X1, X2,......, Xn be a random sample from an exponential population, find the moment estimator for 0.

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(1) Let X = b(16,- find E(4-3x) and distribution function.

B A random process is defined as X (t) A. cos (100 t + 0) where 'A'is a normal random variable with zero mean and unity variance. O is a uniformn random variable over ( n, T) and is independent of A. F'ind the auto correlation function of X (t).

Let X1, . . . , Xn a random sample from a Uniform distribution on the interval [0, 2θ + 1].(a)  Find the MLE of θ.(b) Find the MLE for the variance of the distribution of Xi.

Consider a random sample X1,...,Xn (n > 2) from Beta(θ,1), where we wish to estimate the parameter θ. (a)  Find the MLE θˆ and write it as a function of T = − ∑ni=1 log Xi. (b)  Find the sampling distribution of T = − ∑ni=1 log Xi . (Hint: First find the distribution of Ti = − log Xi .)