Question

Correlation coefficient lies between ______________ and ________________, and probability lies between ______________ and ________________.

• A. $-1$ and $1$, $0$ and $1$
• B. $0$ and $1$, $-1$ and $1$
• C. $-0.5$ and $1.5$, $0$ and $1$
• D. $-1$ and $1$, $-0.5$ and $1.5$

Hint:

Think: \textbf{r in [-1, 1]; P in [0, 1]}. Any option outside these closed intervals is invalid.

Answer 1

The Correct Option is A

Step 1: Correlation bounds. Pearson’s correlation coefficient $r$ satisfies $-1 \le r \le 1$ \Rightarrow lower and upper bounds are $-1$ and $1$.
Step 2: Probability bounds. For any event $A$, probability $P(A)$ obeys $0 \le P(A) \le 1$ \Rightarrow bounds are $0$ and $1$.
Step 3: Eliminate options. (B) swaps the ranges; (C) gives an impossible range for $r$; (D) gives an impossible range for probability. Hence \fbox{(A)} is correct.