Question 1 Let X be a continuous random variable with PDF AutoScroll Print ƒx(x) = { 2² (2x+ 0

Let X be a continuous random variable with PDF 3 x > 1 x4 fx(x) = otherwise Find the mean and variance of x.

Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.

Find the variance of the random variable X with a PDF given by 3x2 S* if - 1< x< 1, f(x) = 2 otherwise. Submit your answer in fractional form.

Show that the variance of a geometric random variable X; is given by V ar(X) = q/p^2 .

Suppose that n observations are chosen at random from a continuous pdf fY(y).  What is the probability that the last observation recorded will be the smallest number in the sample?

The pdf is given as follows: f(x) = { (1/6)e^(1/6x) ; x > 0 0; ew] (i) Find the mean and variance of X using the pdf. (ii) Find the moment generating function of X. (iii) Find the mean and variance of X using the moment generating function.

Please help me with this!! Question 4

At the end of summer, the total weight of seeds accumulated by a nest of seed-gathering ants will vary from nest to nest. If the total weight of seeds accumulated by a nest is exponentially distributed with parameter λ = 1/5, (a) What is the probability that the total combined weight of the seeds gathered by 100 nests will be larger than 4 95 pounds by the end of next summer? (b) What is the probability that the average weight of the seeds gathered by the 100 nests is larger than 5.3 ? (c) What assumptions are you making to answer parts (a) and (b)? Do you think those assumptions make sense in the context of this problem? Explain. (d) Tell us about the possibility that the total combined weight of the seeds gathered by 2 nests follows a normal distribution. Justify your answer in the specific context of this problem (ants, nests, seed-gathering). That is, do not just make generic statements that you think apply to every context in the world.

Derive the variance of the Student's t distribution using the definition Var (x) = ELX2] - EX)² .