Question

   #1 please and the answer should be in the form of a piecewise function

(2n 1)2nn! 2 V2T n=0 Since this is an alternating series (because the parity on the power on x means that will always have the same sign as r), then we can always use the estimates on alternating series which are quite strong to compute values of this sum Problems 1.)Find the probability density function for the random variable representing picking a random real number between -1 and 1. (This is a piecewise defined function.) 2. Find the probability distribution function for the random variable representing picking a random real piecewise defined function.) number tween -1 and 1. (This 3. Compute the mean of the random variable with density function if x >0 if <0. f(x) = 4. Compute the mean of the random variable with density function 2e(1- cos x) if x > 0 if a <O. fx) = 0 5 Compute the variance and standard deviation of the random variable with density function

Answer 1

1.

The probability density function must be positive between -1 and 1.

Also, since it represents picking a number at random, its value must be the same for all numbers

between -1 and 1, then the function is constant in the interval [-1,1]

Therefore the function is of the form

if -1 f(x) kif 1 1 if 1 0

where k is a constant.

Now, recall that the integral of a PDF over all reals must be equal to 1.

Hence

roo kdr 1 f(x)dx

Then

(1)

k(1 (1)) 1

2

Thus, the PDF is

if r 1 if 1 1 if r 0 f (x) 2 0 1