What is the likelihood ratio p(x|C₁) p(x|C₂) in the case of Gaussian densities?

A continuous random variable X has probability distribution function f(x) , wheref(x) = k(1 − x2) for − 1 < x < 1.Find the variance of the random variable X.

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In statistical modelling, it is often the case that you have data obtained from an experiment and a mathematical model that you think is describing the experiment. A likelihood function describes the probability that your experimental data would have occurred as a function of your chosen mathematical model. For example, the likelihood function associated with a "gamma distribution" is given by L(x) = cr-1e Bz where a, B are both strictly positive constants, and the domain of L(x) is (0, 00). 1. Use Product Rule to find the critical numbers of L(x). 2.( 3. ( Use Logarithmic Differentiation to find the critical numbers of L(x). Which method do you prefer? Why? Note: Finding the critical numbers is the first step to maximizing the likelihood function, which makes sense as a next step. You want to find the mathematical model that most likely produced your data!

The velocity of a particle in a gas is a random variable X with probability distribution fx(x) = 32x²e-4x The kinetic energy of the particle is Y = 1/mX². Suppose that the mass of the particle is 64 yg. Find the probability distribution of Y. (Do not convert any units.) x >0.

f (y|0) 3y? exp 03 03 provided y > 0 where 0 > 0 is a parameter. Note that if Y ~ f(y|0), then E(Y³) = 0³. = 0³. (a) Verify that f(y|0) belongs to an exponential family in traditional form. (b) Determine the likelihood function L(0) for this random sample. (c) Show that the maximum likelihood estimator of 0 is 1/3 ÔMLE 3 n i=1 You may use any method that we discussed in class for finding the MLE. (d) Find the Fisher information I(0) in a single observation from this density. (e) Using the standard normal approximation for the distribution of a maximum likelihood estimator based on the Fisher information, construct an approximate 95% confidence interval for 0. Note that z0.025 1.96.

Let x be a continuous random variable with PDF 2x 0 < x < 1 fx(æ) = { otherwise Find the expected value of x.

Find the expected value of the random variable Y whose probability density is given by f( y) =  1/8 (y + 1) for 2 < y < 4 , 0 for elsewhere