Rebekah Robinson, Georgetown College

Always remember to make sure the necessary packages are loaded:

```
require(mosaic)
require(tigerstats)
```

A

continuous random variableis a random variable whose possible values lie in a range of real numbers.

**Examples:**

\( X=\mbox{amount of sugar in an orange.} \)

- Range of \( X \) = \( (0,\infty) \).

\( Y=\mbox{heights of students at GC} \)

- Range of \( Y=(0,\infty) \).

\( Z=\mbox{student grades in MAT 111} (\%) \)

- Range of \( Z=[0,100] \)

The PDF for a *discrete* random variable is a table or a histogram.

The PDF for a *continuous* random variable is a smooth curve.

PDF for a **normal** random variable \( X \):

**Normal curve**Bell-shaped

Symmetric about \( EV(X)=\mu \)

\( SD(X)=\sigma \)

**68-95**Rule applies

- Probability that \( X \) falls within 1 standard deviation of the mean:

\[ P(\mu-\sigma< X < \mu+\sigma)\approx 0.68 \]

- Probability that \( X \) falls within 2 standard deviations of the mean:

\[ P(\mu-2\sigma< X < \mu+2\sigma)\approx 0.95 \]

- Probability that \( X \) falls within 3 standard deviations of the mean:

\[ P(\mu-3\sigma< X < \mu+3\sigma)\approx 0.997 \]

**Example:** The heights of college males follow a normal distribution with \( \mu=72 \) inches and \( \sigma=3.1 \) inches. Let

\[ X= \mbox{ height of a college male.} \]

About 95% of college males are between what two heights?

\( P( \) ________ < \( X \) < ________\( ) \) \( \approx \) 0.95

```
require(manipulate)
EmpRuleGC(mean=72,sd=3.1)
```

Calculating other probabilities for \( X \) is easy.

To find \( P(X>70.9) \):

```
pnormGC(70.9,region="above",mean=72,
sd=3.1,graph=TRUE)
```

```
[1] 0.6386
```

\[ P(X>70.9)=0.6386 \]

To find \( P(X<69.4 \mbox{ or } X>79.1) \):

```
pnormGC(c(69.4,79.1),region="outside",
mean=72,sd=3.1,graph=TRUE)
```

```
[1] 0.2118
```

\[ P(X<69.4 \mbox{ or } X>79.1) \] \[ =0.2118 \]

Suppose we want to know the height of a male that is taller than 80% of college men.

\[ P(X\leq x)=0.80 \]

To find the **percentile ranking**, \( x \)

```
qnorm(0.80,mean=72,sd=3.1)
```

```
[1] 74.61
```

A 74.61 inch tall male is taller than 80% of college men.

A normal distribution is a good approximation to a binomial distribution if:

there are at least 10 expected successes, \( np\geq 10 \) and

there are at least 10 expected failures, \( n(1-p)\geq 10 \).

```
require(manipulate)
BinomNorm()
```

If the expected number of success or failures is too small, this approximation is not very good.

```
require(manipulate)
BinomSkew()
```