The normal distribution curve, which models the distributions of data in a wide range of applications, is given by the function $p ( x ) = \frac { 1 } { \sqrt { 2 \pi } \sigma } e ^ { \frac { - ( x - \mu ) ^ { 2 } } { 2 \sigma ^ { 2 } } }$ , $\pi = 3.14159265 \ldots$ and and are constants called the standard deviation and the mean, respectively. With $\sigma = 3$ and $\mu = 5$ , approximate $\int _ { - \infty } ^ { \infty } p ( x ) \mathrm { d } x$ .
A)3
B) -5
C) 5
D) -1
E) 1

Question

The normal distribution curve, which models the distributions of data in a wide range of applications, is given by the function $p ( x ) = \frac { 1 } { \sqrt { 2 \pi } \sigma } e ^ { \frac { - ( x - \mu ) ^ { 2 } } { 2 \sigma ^ { 2 } } }$ , $\pi = 3.14159265 \ldots$ and   and   are constants called the standard deviation and the mean, respectively. With $\sigma = 3$ and $\mu = 5$ , approximate $\int _ { - \infty } ^ { \infty } p ( x ) \mathrm { d } x$ .  
A)3
B) -5
C) 5
D) -1
E) 1

Quizplus · Accepted Answer

The integral of the normal distribution function over its entire range (from $-\infty$ to $\infty$) is always equal to 1. This is because the function represents a probability distribution, and the total area under the curve must sum to 1 to represent the total probability space.

The Normal Distribution Curve, Which Models the Distributions of Data p(x)=12πσe−(x−μ)22σ2p ( x ) = \frac { 1 } { \sqrt { 2 \pi } \sigma } e ^ { \frac { - ( x - \mu ) ^ { 2 } } { 2 \sigma ^ { 2 } } }p(x)=2π​σ1​e2σ2−(x−μ)2​

The Normal Distribution Curve, Which Models the Distributions of Data $p ( x ) = \frac { 1 } { \sqrt { 2 \pi } \sigma } e ^ { \frac { - ( x - \mu ) ^ { 2 } } { 2 \sigma ^ { 2 } } }$