Read normal parameters and describe their graph effect

Which parameter reading is correct for this density?

Read the same model in different forms

A normal density has a real mean and a positive standard deviation. The leading factor and the exponent describe the same parameters, so either can be used to check your reading

1. 1Read the centre Match x − μ. A term x + c has mean −c. \(\displaystyle x+5=x-(-5)\;\Rightarrow\;\mu=-5\)
2. 2Rearrange the exponent coefficient If the coefficient is b, take its reciprocal, divide by 2, then take the positive square root for the standard deviation \(\displaystyle \frac{1}{2\sigma^2}=b\;\Rightarrow\;\sigma^2=\frac{1}{2b},\quad\sigma>0\)
3. 3Distinguish variance from standard deviation In normal-distribution notation the second parameter is the variance. The front coefficient is 1 divided by σ√(2π) \(\displaystyle X\sim N(\mu,\sigma^2)\)
4. 4Use the complete normal model The curve is symmetric about its mean, which is also its median and mode. The density is positive at every finite real value, its tails approach but never meet the horizontal axis, and its total area is 1. \(\displaystyle f(\mu-d)=f(\mu+d),\quad\int_{-\infty}^{\infty}f(x)\,dx=1\)
5. 5Compare centre and spread separately Changing only the mean translates the curve. A larger standard deviation makes it wider with a lower peak; its total area remains 1. \(\displaystyle f(\mu)=\frac{1}{\sigma\sqrt{2\pi}}\)

Model

\(\displaystyle f(x)=\frac{1}{3\sqrt{2\pi}}\exp\left(-\frac{1}{18}(x-7)^2\right)\)

Read the mean from this density

Model

\(\displaystyle f(x)=\frac{1}{3\sqrt{2\pi}}\exp\left(-\frac{1}{18}(x-7)^2\right)\)

Find the variance from this density

The squared centred term in a normal density is (x + 5)². Which mean is correct?

Model

\(\displaystyle f(x)=\frac{2}{3\sqrt{2\pi}}\exp\left(-\frac{2}{9}(x+2)^2\right)\)

Find the variance. Give an exact fraction or decimal

Normal model

\(\displaystyle X\sim N(-1,4)\)

For this continuous normal variable, what does the height at x = −1 represent? The plotted window shows only part of the curve

Normal model

\(\displaystyle X\sim N(-1,4)\)

Which statement is false for this normal model?

Compare centre and spread separately

Read both standard deviations before comparing widths. With the mean fixed, a larger standard deviation spreads the same total area over a wider curve and lowers its peak. Changing only the mean moves the curve without changing its width or height

1. 1Read variance notation carefully The second parameter is σ², so take its positive square root before comparing standard deviations \(\displaystyle X_A\sim N(-1,4),\quad X_B\sim N(2,9)\;\Rightarrow\;\sigma_A=2,\quad\sigma_B=3\)
2. 2Compare peak heights The reciprocal relationship means the larger standard deviation has the lower peak \(\displaystyle \frac{f_B(2)}{f_A(-1)}=\frac{2}{3}\)

Models

\(\displaystyle X_A\sim N(-1,4),\quad X_B\sim N(2,9)\)

How does Curve B differ from Curve A? Both are normal densities; solid A and dashed B continue beyond the viewing window

Model

\(\displaystyle X_A\sim N(7,4)\)

Curve B has a mean 3 units smaller than Curve A and a standard deviation one half unit smaller. Find the variance of Curve B.

Use the parameters to calculate moments

The mean and variance determine the second moment. For a linear transformation, apply the multiplier to the mean but its square to the variance; an added constant does not change variance

1. 1Mean and variance Read these directly from the model notation \(\displaystyle E(X)=\mu,\quad\operatorname{Var}(X)=\sigma^2\)
2. 2Second moment Rearrange the variance identity. Do not confuse E(X²) with [E(X)]² \(\displaystyle E(X^2)=\sigma^2+\mu^2\)
3. 3Linear expectation Multiply the mean by a and then add b \(\displaystyle E(aX+b)=a\mu+b\)
4. 4Linear variance Square a and multiply the original variance. A constant shift does not affect spread \(\displaystyle \operatorname{Var}(aX+b)=a^2\sigma^2\)

Model

\(\displaystyle X\sim N(-3,4)\)

Find E(X²) for this normal variable

Model

\(\displaystyle X\sim N(-3,4)\)

Find E(−2X + 1)

Model

\(\displaystyle X\sim N(-3,4)\)

Find Var(−2X + 1)

Normal model

\(\displaystyle X\sim N(-3,4)\)

For this model, which calculation correctly distinguishes the second moment from the squared mean?