A uniform random variable X has a probability density given by: 1/0≤x≤2 otherwise f(x) = The mean of X is 1 and variance is 1/3. (a) Find the probability that X takes values within 2 standard deviations of the mean i.e. P(|X−1| < 20). (b) Find a bound for the probability in part (a) using the Chebychev's inequality.

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**Uniform Random Variable Analysis**

A uniform random variable \( X \) has a probability density given by:

\[ 
f(x) = 
\begin{cases} 
\frac{1}{2} & \text{for } 0 \leq x \leq 2 \\ 
0 & \text{otherwise} 
\end{cases} 
\]

The **mean** of \( X \) is 1, and the **variance** is \( 1/3 \).

**Tasks**

(a) Find the probability that \( X \) takes values within 2 standard deviations of the mean, i.e., \( P(|X - 1| < 2\sigma) \).

(b) Find a bound for the probability in part (a) using the Chebychev's inequality.
Transcribed Image Text:**Uniform Random Variable Analysis** A uniform random variable \( X \) has a probability density given by: \[ f(x) = \begin{cases} \frac{1}{2} & \text{for } 0 \leq x \leq 2 \\ 0 & \text{otherwise} \end{cases} \] The **mean** of \( X \) is 1, and the **variance** is \( 1/3 \). **Tasks** (a) Find the probability that \( X \) takes values within 2 standard deviations of the mean, i.e., \( P(|X - 1| < 2\sigma) \). (b) Find a bound for the probability in part (a) using the Chebychev's inequality.
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