Question

which of the following are properties of the normal density curve?
select all that apply.
a. as the value of x increases, the graph approaches, and eventually equals, zero. when the value of x decreases, the graph approaches, and eventually equals, zero
b. its highest point occurs at μ.
c. it is symmetric about its mean μ.
d. the area under the curve to the right of μ equals the area under the curve to the left of μ. both equal 1/2
e. the area under the curve is 1.

Explanation:

• Option A: The normal density curve approaches \(0\) as \(x\to\pm\infty\), but it never actually equals \(0\). So, this option is incorrect.
• Option B: The normal density curve \(y = f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x - \mu)^{2}}{2\sigma^{2}}}\). Taking the first - derivative \(y^\prime=-\frac{x-\mu}{\sigma^{3}\sqrt{2\pi}}e^{-\frac{(x - \mu)^{2}}{2\sigma^{2}}}\) and setting \(y^\prime = 0\), we get \(x=\mu\). The second - derivative \(y^{\prime\prime}=\frac{(x - \mu)^{2}-\sigma^{2}}{\sigma^{5}\sqrt{2\pi}}e^{-\frac{(x - \mu)^{2}}{2\sigma^{2}}}\), and at \(x = \mu\), \(y^{\prime\prime}<0\). So, the maximum of the function occurs at \(x=\mu\). This option is correct.
• Option C: The normal density function \(f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x - \mu)^{2}}{2\sigma^{2}}}\), and \(f(\mu + a)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{a^{2}}{2\sigma^{2}}}\), \(f(\mu - a)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(-a)^{2}}{2\sigma^{2}}}\). Since \(f(\mu + a)=f(\mu - a)\), the curve is symmetric about \(x = \mu\). This option is correct.
• Option D: Because of the symmetry about \(x=\mu\), the area under the curve \(A=\int_{-\infty}^{\infty}f(x)dx = 1\), and \(\int_{-\infty}^{\mu}f(x)dx=\int_{\mu}^{\infty}f(x)dx=\frac{1}{2}\). This option is correct.
• Option E: The total area under any probability density function \(y = f(x)\) is \(A=\int_{-\infty}^{\infty}f(x)dx = 1\). For the normal density function \(f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x - \mu)^{2}}{2\sigma^{2}}}\), \(\int_{-\infty}^{\infty}\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x - \mu)^{2}}{2\sigma^{2}}}dx = 1\). This option is correct.

Answer:

B. Its highest point occurs at \(\mu\); C. It is symmetric about its mean \(\mu\); D. The area under the curve to the right of \(\mu\) equals the area under the curve to the left of \(\mu\). Both equal \(\frac{1}{2}\); E. The area under the curve is \(1\)