First slide

Introduction to probability

Question

A random variable X takes the values – 2, -1, 1 and 2 with probabilities $\frac{1 - a}{4}, \frac{1 + 2 a}{4}, \frac{1 - 2 a}{4}$ and $\frac{1 + a}{4}$ respectively then

Moderate

A

$a c a n h a v e a n y r e a l v a l u e$
B

$- \frac{1}{2} \leq a \leq \frac{1}{2}$
C

$- 1 \leq a \leq 1$
D

$\frac{1}{4} \leq a \leq \frac{1}{3}$

Solution

$\begin{array}{l} 0 \leq \frac{1 - a}{4} \leq 1 \\ \Rightarrow 0 \leq 1 - a \leq 4 \\ \Rightarrow 0 - 1 \leq - a \leq 3 \\ \Rightarrow - 3 \leq a \leq 1 \end{array}$
$\begin{array}{l} 0 \leq \frac{1 + 2 a}{4} \leq 1 \\ \Rightarrow 0 \leq 1 + 2 a \leq 4 \\ \Rightarrow - 1 \leq 2 a \leq 3 \\ \Rightarrow \frac{- 1}{2} \leq a \leq \frac{3}{2} \end{array}$
$\begin{array}{l} 0 \leq \frac{1 - 2 a}{4} \leq 1 \\ \Rightarrow 0 \leq 1 - 2 a \leq 4 \\ \Rightarrow - 1 \leq - 2 a \leq 3 \\ \Rightarrow - \frac{3}{2} \leq a \leq \frac{1}{2} \end{array}$
$\begin{array}{l} 0 \leq \frac{1 + a}{4} \leq 1 \\ 0 \leq 1 + a \leq 4 \\ 0 \leq a \leq 3 \\ therefore a \in [\frac{- 1}{2}, \frac{1}{2}] \end{array}$

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