(25 pts) Consider a random variable X with density function f X (x) = x , 0 ≤ x ≤1 0, else ⎧ ⎨ ⎪ ⎩ ⎪ . a. (3) Carefully sketch by hand the density function f X (x) . Be sure to dimension both axes. b. (4) Find the mean and variance of X . c. (4) Suppose we generate a new random variable Y = Q(X) by quantizing X according to the following 3-level uniform quantizer: Q(x) = − 2 3 , −1≤ x < − 1 3 0, − 1 3 ≤ x < 1 3 2 3 , 1 3 ≤ x ≤1

Step-by-step explanation:

It seems that this question is incomplete, and unfortunately, no reference was found in the internet. However, it seems that question is about calculating the variance and the mean of a random variable, based on its pdf (probability density funtion). Recall that a function must fulfill the following property for it to be a pdf

Also, recall that the following formulas

(the mean)

.

Let us illustrate this calculations with an example.

Consider the function f(x) = 2x if and 0 otherwise. By easy calculations, we can check that f(x) is indeed a pdf (it integrates up to 1). Hence it's mean is

and the variance is given by

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step-by-step explanation: