Question

Given the random variable X in Problem 2.4.2, let V=g(X)=|X|.

(a) Find Pᵥ(v).
(b) Find Fᵥ(v).
(c) Find E[V].

2.4.2 The random variable X has CDF
Fx (x) = {0 x < -1,
{0.2 -1 ≤ x < 0,
{0.7 0 ≤ x < 1,
{1 x ≥ 1.

Answer 1

The question involves finding the probability distribution function (PDF), cumulative distribution function (CDF), and expected value (E[V]) for the transformed random variable V, where V = |X|, given the explicit CDF of X. Calculations require integrating the probabilities over the range of V and using the symmetry of the distribution of X.

Step-by-step explanation:

The random variable X is defined with a piecewise cumulative distribution function (CDF) as provided. Given V = |X|, we aim to find the corresponding probability distribution function (PDF), cumulative distribution function (CDF), and the expected value (E[V]) of V.

Part a: Find PV(v)

To find the PDF of V, we note that V takes on the values of X without regard to sign. Since X takes on values within [-1,1], V will take on values within [0,1]. By considering the symmetry of X around 0 and the definition of V, we can deduce the probability mass at each point of V from the provided CDF of X.

We find P(V = v) = P(X = v) + P(X = -v) for v in the range [0, 1]. Using the given CDF of X, we get PV(v) = FX(v) - FX(-v-), where '-' denotes the left limit.

Part b: Find FV(v)

The CDF of V, FV(v), can be obtained by integrating PV(v) over the range from 0 to v. We can compute this by considering the contributions from the positive and negative parts of X.

Part c: Find E[V]

The expected value of V, E[V], is the integral of vPV(v) from 0 to 1. This is computed by integrating the product of the value and its probability, taking into account the symmetry of the distribution of X and, hence, V.