GATE EC 2022 Set 1 Q54 — Digital Communications

Consider a real valued source whose samples are independent and identically distributed random variables with the probability density function, $f(x),$ as shown in the figure.

Consider a $1$ bit quantizer that maps positive samples to value $\alpha$ and others to value $\beta.$ If $\alpha^{\ast}$ and $\beta^{\ast}$ are the respective choices for $\alpha$ and $\beta$ that minimize the mean square quantization error, then $(\alpha^{\ast} – \beta^{\ast}) =$____________ (rounded off to two decimal places).