This question was previously asked in

GATE PI 2016 Official Paper

Option 2 : \(\frac{1}{2}\)

**Concept:**

Normal/Gaussian/Bell distribution:

Probability distribution function (PDF) for a normal distribution is:

\(PDF = f\left( x \right) = \frac{1}{{\sqrt {2\pi {σ ^2}\;} }}{e^{ - \frac{1}{2}{{\left( {\frac{{x - μ }}{σ }} \right)}^2}}}\)

where

x = normal random variable

μ = mean = mode = median

σ = standard deviation and σ^{2} = variance.

Note:

**1) Normal distribution is symmetric about its mean.**

**Calculation:**

**Given:**

\(f(x)=\frac{1}{\sqrt{8\pi}}e^{-\left \{ \frac{(x\;-\;1)^2}{8} \right \}}\)

Comparing it with the standard normal distribution

μ = mean = 1 and σ^{2} = variance = 4

The distribution function is divided into two equal parts which are equiprobable.

\(\therefore\; \int_{-\infty}^{\infty}f(x)dx=1\)

\(\therefore\; \int_{-\infty}^{1}f(x)dx\;+\;\int_{1}^{\infty}f(x)dx=1\)

∵ A normal distribution is symmetric about mean i.e. \(\therefore\; \int_{-\infty}^{1}f(x)dx\;=\;\int_{1}^{\infty}f(x)dx\)

\(\therefore\; \;\int_{1}^{\infty}f(x)dx=\frac{1}{2}\)