Consider X follows uniform distribution with A =-5 and B=7. Calculate P(−2.5≤X≤2.5) 𝑃 ( − 2.5 ≤ 𝑋 ≤ 2.5 ) a. 0.56 b. 0.24 c. 0.42 d. 0.30

1. Break Down the Problem

We need to calculate the probability $P(-2.5 \leq X \leq 2.5)$ for a uniform distribution defined by the interval $[-5, 7]$.

2. Relevant Concepts

For a uniform distribution defined by $A$ and $B$:

• The probability density function (PDF) is given by: $f(x) = \frac{1}{B - A} \quad \text{for } A \leq x \leq B$
• The probability of $X$ falling within an interval $[c, d]$ is given by: $P(c \leq X \leq d) = (d - c) \cdot f(x)$

3. Analysis and Detail

In our case:

• $A = -5$
• $B = 7$
• Calculating $B - A$: $B - A = 7 - (-5) = 12$
• The PDF $f(x)$ is: $f(x) = \frac{1}{12}$
• We want to find $P(-2.5 \leq X \leq 2.5)$:
  • Calculate the width of the interval: $d - c = 2.5 - (-2.5) = 2.5 + 2.5 = 5$
• Now, we can find the probability: $P(-2.5 \leq X \leq 2.5) = (2.5 - (-2.5)) \cdot f(x) = 5 \cdot \frac{1}{12} = \frac{5}{12}$

4. Verify and Summarize

Converting $\frac{5}{12}$ to a decimal gives: $\frac{5}{12} \approx 0.4167$

Final Answer

The closest answer choice is: c. 0.42