find the expected and variance value of RV x

f(x)= 
⎩
⎨
⎧
​
  
2
1
​
 x
0.75
0
​
  
if 0<x<1
if 2<x<3
otherwise
​


To find the expected value \( E(X) \) and variance \( \text{Var}(X) \) of the random variable \( X \) given its probability density function (pdf) \( f(x) \), we can proceed as follows:

Given:
\[
f(x) = 
\begin{cases}
2, & \text{if } 0 < x < 1 \\
\frac{1}{4}, & \text{if } 2 < x < 3 \\
0, & \text{otherwise}
\end{cases}
\]

### Step 1: Check that \( f(x) \) is a valid pdf
For \( f(x) \) to be a valid probability density function, the total area under \( f(x) \) over all \( x \) must equal 1.

#### Integrate over the two intervals:
\[
\int_0^1 2 \, dx = 2 \cdot (1 - 0) = 2
\]
\[
\int_2^3 \frac{1}{4} \, dx = \frac{1}{4} \cdot (3 - 2) = \frac{1}{4}
\]
These areas add up to \( 2 + \frac{1}{4} = 2.25 \), which does not equal 1. Therefore, \( f(x) \) as given is not a valid pdf, as its total area is not normalized to 1.

For \( f(x) \) to be used as a probability density function, it must be scaled to ensure that the total probability sums to 1.

Find the expected and variance value of RV x where f(x) = { 2 if 0 < x < 1, 1/4 if 2 < x < 3, and 0 otherwise.}

This page explains how to calculate the expected value and variance of a random variable X, given its probability density function f(x) defined piecewise. The solution includes checking if f(x) is a valid PDF and using integration techniques to find E(X) and Var(X). Suitable for undergraduate-level students studying probability.

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