Griffiths D. Head First Statistics

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using the normal distribution ii

Beyond Normal

If only all probability distributions were normal.

Life can be so much simpler with the normal distribution. Why spend all your time

working out individual probablities when you can look up entire ranges in one swoop, and

still leave time for game play? In this chapter, you’ll see how to solve more complex

problems in the blink of an eye, and you’ll also find out how to bring some of that normal

goodness to other probability distributions.

If only all probability distributions were normal.

Life can be so much simpler with the normal distribution. Why spend all your time

working out individual probablities when you can look up entire ranges in one swoop, and

still leave time for game play? In this chapter, you’ll see how to solve more complex

problems in the blink of an eye, and you’ll also find out how to bring some of that normal

goodness to other probability distributions.

362 Chapter 9

Love is a roller coaster

The wedding market is big business nowadays, and Dexter has an idea for

making that special day truly memorable. Why get married on the ground when

you can get married on a roller coaster?

Dexter’s convinced there’s a lot of money to be made from his innovative Love

Train ride, if only it passes the health and safety regulations.

Before Dexter can go any further, he needs to make

sure that his special ride can cope with the weight

of the bride and groom, and he’s asked if you can

help him.

The ride he has in mind can cope with combined weights of up to 380

pounds. What’s the probability that the combined weight will be less

than this?

I need to make sure

the combined weight of

the bride and groom won’t

be above 380 pounds.

Think you can help?

I dooooOOOooooo!

Dexter

come on, ride the love train

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using the normal distribution ii

All aboard the Love Train

Before we start, we need to know how the weights of brides and grooms in

Statsville are distributed, taking into account the weight of all their wedding

clothes. Both follow a normal distribution, with the bride weight distributed

as N(150, 400) and the groom weight as N(190, 500). Their weights are

measured in pounds.

How do you think we can find the probability

distribution for the combined weights of the bride and

groom? What sort of distribution do you think this

might be? Why?

We need to use these two probability distributions to somehow work out

the probability that the weight of a bride and groom will be less than the

maximum weight allowance on the ride. If the probability is sufficiently

high, we can be confident the ride is feasible.

We can calculate

this probability if we

know what the combined

probability distribution is,

but what’s that?

Bride ~ N(150, 400)

150

Groom ~ N(190, 500)

190

364 Chapter 9

Normal bride + normal groom

Let’s start by taking a closer look at how the weights of the bride and groom

are distriuted.

As you know, the weights follow normal distributions like this:

Most brides will have

a weight around here

And most grooms

will have a weight

close to here.

Brides can be lighter or

heavier than the mean, and

this applies to the grooms too.

What we’re really after, though, is the probability distribution of the

combined weight of the bride and groom. In other words, we want to find

the probability distribution of the weight of the bride added to the weight

of the groom.

Bride weight + Groom weight ~ ?

Assuming the weights of the bride and groom are independent, the

shape of the distribution should look something like this:

Most couples will have

a joint weight around

here.

There’s a lot of variation in

combined weight, though.

150

190

Bride μ Groom μ

Bride + Groom μ

adding two normal distributions

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using the normal distribution ii

It’s time for a trip down memory lane. Can you remember the

discrete shortcuts for the following formulas? Assume X and Y are

independent.

1. E(X + Y)

3. E(X - Y)

2. Var(X + Y)

4. Var(X - Y)

It’s still just weight

Can you remember when we first looked at continuous data and looked at how data

such as height and weight tend to be distributed? We found that data such as height

and weight are continuous, and they also tend to follow a normal distribution.

This time we’re looking at the combined weight of the happy couple. Even though

it’s combined weight, it’s still just weight, and we already know how weight tends to be

distributed. The combined weight is still continuous. What’s more, the combined

weight is still distributed normally. In other words, the combined weight of the

bride and groom follows a normal distribution.

Knowing that the combined weight of the bride and groom follows a normal

distribution helps us a lot. It means that we’ll be able to use probability tables just

like we did before to look up probabilities, which means we’ll be able to look up the

probability that the combined weight is less than 380 pounds—just what we need for

the ride.

There’s only one problem—before we can go any further, we need to know the mean

and variance of the combined weight of the bride and groom. How can we find this?

Bride weight + Groom weight ~ N(?, ?)

The combined weight

of the bride and

groom follows a normal

distribution, but what’s

the mean and variance?

366 Chapter 9

It’s time for a trip down memory lane. Can you remember the

discrete shortcuts for the following formulas? Assume X and Y are

independent.

I don’t see how these

shortcuts help us. They’re

for discrete data, and we’re

dealing with continuous now.

The shortcuts apply to continuous data too.

When we originally encountered these shortcuts, we were dealing with discrete data.

Fortunately, the same rules and shortcuts also apply to continuous data.

1. E(X + Y)

3. E(X - Y)

2. Var(X + Y)

4. Var(X - Y)

E(X + Y) = E(X) + E(Y)

E(X - Y) = E(X) - E(Y)

Var(X + Y) = Var(X) + Var(Y)

Remember that we

ADD the variances, even

though it’s for X - Y.

How do you think we can use these shortcuts to find the probability

distribution of the weight of the bride + the weight of the groom?

sharpen solution

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using the normal distribution ii

How’s the combined weight distributed?

So far, we’ve found that the combined weight of the bride and groom are

normally distributed, and this means we can use probability tables to look up

the probability of the combined weight being less than a certain amount.

Let’s try rewriting the bride and groom weight distributions in terms of X

and Y. If X represents the weight of the bride and Y the weight of the groom,

and X and Y are independent, then we want to find μ and σ where

X + Y ~ N(μ, σ

)

X + Y means “the weight of the bride + the

weight of the groom.” But how do we know

what the mean and variance are?

In other words, before we go any further we need to find the mean and variance of

X + Y. But how?

Take a look at the answers to the last exercise. When we were working with discrete

probability distributions, we saw that as long as X and Y are independent we could

work out E(X + Y) and Var(X + Y) by using

E(X + Y) = E(X) + E(Y) and Var(X + Y) = Var(X) + Var(Y)

So if we know what the expectation and variance of X and Y are, we can use these

to work out the expectation and variance of X + Y.

That means that if we

know the distribution of X

and Y, we can figure out the

distribution of X + Y too.

We can use what we already know to figure out

what we don’t.

Because we know how the weight of the bride and the weight of the

groom are distributed, we can find the distribution of the combined

weight of the bride and groom.

Let’s look at this in more detail.

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X + Y Distribution Up Close

Being able to find the distribution of X + Y is useful if you’re working

with combinations of normal variables. If independent random variables

X and Y are normally distributed, then X + Y is normal too. What’s

more, you can use the mean and variance of X and Y to calculate the

distribution of X + Y.

To find the mean and variance of X + Y, you can use the same formulae

that we used for discrete probability distributions. In other words, if

X ~ N(μ

, σ

) and Y ~ N(μ

, σ

)

then

where

μ = μ

+ μ

= σ

+ σ

X + Y ~ N(μ, σ

)

If you add the means of X and Y together,

you get the mean of X + Y. Similarly,

summing the variances of X and Y gives you

the variance of X + Y

In other words, the mean of X + Y is equal to the mean of X plus the

mean of Y, and the variance of X + Y is equal to the variance of X plus

the variance of Y.

Let’s look at a sketch of this. What do you notice about the variance of

X + Y?

The variance of X + Y is greater than the variance of X and also

greater than the variance of Y, which means that the curve of X + Y

is more elongated than either. This is true for any normal X and Y.

By adding the two variables together, you are in effect increasing the

amount of variability, and this elongates the shape of the distribution.

This in turn means that the shape of the distribution gets flatter so that

the total area under the curve is still 1.

Remember, two variables are

independent if they have no impact

on each other’s probabilities.

We can use these shortcuts if

X and Y are independent, which

makes life very easy indeed

X ~ N(μ

, σ

)

X + Y ~ N(μ

, σ

+ σ

)

Y ~ N(μ

, σ

)

x + y and x – y in depth

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using the normal distribution ii

where

Sometimes X + Y just won’t give you the sorts of probabilities you’re

after. If you need to find probabilities involving the difference between two

variables, you’ll need to use X - Y instead.

X - Y follows a normal distribution if X and Y are independent random

variables and are both normally distributed. This is exactly the same

criteria as for X + Y.

To find the mean and variance, we again use the same shortcuts that we

used for discrete probability distributions. If

X ~ N(μ

, σ

) and Y ~ N(μ

, σ

)

then

μ = μ

- μ

= σ

+ σ

X - Y ~ N(μ, σ

)

In other words, the mean of X – Y is equal to the mean of Y subtracted

from the mean of X, and you find the variance of X – Y by adding the X

and Y variances together.

We ADD the variances together,

just like we did for discrete

probability distributions.

X ~ N(μ

, σ

)

X - Y ~ N(μ

, σ

+ σ

)

Y ~ N(μ

, σ

)

– =

Subtract the mean,

add the variance.

Adding the variances together may not make intuitive sense at first,

but it’s exactly the same as when we worked with discrete probability

distributions. Even though we’re subtracting Y from X, we’re actually

still increasing the amount of variability. Adding the variances together

reflects this. As with the X + Y distribution, this leads to a flatter, more

elongated shape than either X or Y

If you look at the actual shape of the X - Y distribution, it’s the same

shape curve as for X + Y distribution, except that the center has moved.

The two distributions have the same variances, but different means.

Look at the shape. It’s the same

shape as for X + Y but with the

center of the curve in a different

place. They have the same shape

because they have the same variance.

X – Y Distribution Up Close

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Remind me, why did we need to

find the distribution of X + Y?

A: We’re looking for the probability that

the combined weight of a bride and groom

will be less than 380 pounds, which means

we need to know how the combined weight

is distributed. We’re using X to represent the

weight of the bride, and Y to represent the

weight of the groom, which means we need

to use the distribution of X + Y.

You say we can look up

probabilities for X + Y using probability

tables. How?

A: In exactly the same way as we did

before. We take our probability distribution,

calculate the standard score, and then look

this value up in probablity tables.

Looking up probabilities for X + Y is no

different from looking up probabilities for

anything else. Just find the standard score,

look it up, and that gives you your probability.

So do all of the shortcuts we

learned for discrete data apply to

continuous data too?

A: Yes, they do. This means we have an

easy way of combining random variables

and finding out how they’re distributed,

which in turn means we can solve more

complex problems.

The key thing to remember is that these

shortcuts apply as long as the random

variables are independent.

Can you remind me what

independent means?

A: If two variables are independent, then

their probabilities are not affected by each

other. In our case, we’re assuming that the

weight of the bride is not influenced by the

weight of the groom.

What if X and Y aren’t independent?

What then?

A: If X and Y aren’t independent, then

we can’t use these shortcuts. We’d need to

do a lot more work to find out how X + Y is

distributed because you’d have to find out

what the relationship is between X and Y.

Finding probabilities

Now that we know how to calculate the distribution of X + Y, we

can look at how to use it to calculate probabilities. Here are the

steps you need to go through.

Work out the distribution and range

Standardize it

Look up the probabilities

Sound familiar? These are exactly the same steps that

we went through in the previous chapter for the normal

distribution.

We know we need to use X + Y, and

we have a way of working out the

mean and variance.

Once we know the distribution

and the range, we standardize it.

We can then look up the

probability in standard

normal probability tables.

calculating probabilities for x + y