STA220 - ENVIRONMENTAL SCIENCES

Oct. 5

TODAY’S CLASS

• Muddiest points

• The Normal probability distributions

• The Binomial probability distributions

Binomial :

1 .

n independent trials , n IS Fixed

2>

Each trial has a p rob. p

of Success and t P of failure

p is the same for each trial.

then the total # of successesIn n trims has a Bin ( n , p )

90% of Can and ray have hadchrcvun pox by the time theyare adults .

A random Sample of 1 5 Can . antis tavun .

What is the probabilitythat exactly hohad oh run pox ?

X = total # of cases with Chunn PXBin ( 15 ,

o . 9) p (X=to ) = ?

Binds ,0.9 )

PHIe-5)YNBM ( 15,0 . D

p#wT (,

pK=: -0105

l-2. - -

- - . .. .

. -

is

10 Successes . Same as 5 failures

Binds, oil ) instead of

Bin ( Is ,o.9 )p*-n)=€Dp*(tpyw '

Xls Bin 15,09 )

✓

Sally gets a cup of coffee and a mun every day for breakfast from one of the many coffee shops in her neighbourhood. She picks a coffee shop each morning at random and independently of previous days. The average price of a cup of coffee is $1.40 with a standard deviation of $0.30, the average price of a muffin is $2.50 with a standard deviation of $0.15, and the two prices are independent of each other.

What is the mean and standard deviation of the amount she spends on breakfast for two days.

tan

Bren = Cost + mv # .

E $nerd = F- ( o of +Murff= F- (Co # ) t E ( mu A)= 1 . 4 of 2 . So =3 Mo

Van ( Bren ) = Var @ It muF)= V nr (Co Hee ) + Vw rxvif )

= . 3 it . 152= - 1125

2 days .

E Hren+ Benz ) = E Areva) t F- Bland=

2×33.90Var ( Brent Ben ) = Var ( BenDTVAHDEKYSD = FIX = o.gg

= . 1125 to1125=24. 1125

NORMAL DISTRIBUTION • Normal distribution is an idealized mathematical model for some distributions of real data.

•Developed originally by A. De Moivre (1733), but named after C. F. Gauss.

THE NORMAL DISTRIBUTION

Nuno )/ |

THE 68-95-99.7 RULEAll normal distributions have the following property:

68% of area under curve lies within σ of the mean

95% of area under curve lies within 2σ of the mean

99.7% of area under curve lies within 3σ of the mean

THE 68-95-99.7 RULE

IS NORMAL NORMAL?Historically, considerable effort was devoted to proving (unsuccessfully) that all variables follow a normal distribution.

“…the Law of Error upon which these Normal Values are based…finds a footing wherever the individual peculiarities are wholly due to the combined influence of a multitude of accidents…”

Francis Galton, Natural Inheritance, 1889, page 54-55

APPLICATIONS OF THE NORMAL DISTRIBUTION

• Normal distributions are good approximations for the distributions of many variables (e.g., weight, height, IQ, blood pressure, cholesterol level).

• However, many variables do not follow normal distributions (counts, income, expenditure).

Area to the left of -1 = Probability that X is less than -1, denoted

Pr (X < -1). X represents a random quantity (e.g., celsius temperature)

NORMAL DISTRIBUTIONS• If Z has a N(0,1) distribution, then

• σZ is N(0,σ),

• Y = µ + σZ is N(µ,σ) Linear Transformation of Z

• Conversely, if Y is N(µ,σ) distributed, the

• Z=(Y- µ)/σ has a N(0,1) distribution

• The last relationship leads to standardization and the use of standard normal distribution (N(0,1)) to compute relative frequencies

NORMAL DISTRIBUTION

Properties:

• All normal distributions have same bell shape

• Differ in their centre and spread

• Centre: Mean (denoted by µ, Greek letter mu)

• Spread: SD (denoted by σ, Greek letter sigma)

NORMAL DISTRIBUTIONS

Understand how to use these tables … it is posted on the course website … this table will be provided on midterm and final exams.

.*¥t¥y #:*

P(zEtl2).

=0.8686

pftzrhfl - k686

EXAMPLE: DEAR ABBY…Dear Abby: You wrote in your column that a woman is pregnant for 266 days. Who said so? I carried my baby for 10 months and 5 days, and there is no doubt about it because I know the exact date my baby was conceived. My husband is in the Navy and it couldn't have possibly been conceived any other time because I saw him only once for an hour, and I didn’t see him again until the day before the baby was born. I don’t drink or run around, and there is no way the baby isn’t his, so please print a retraction about the 266-day carrying time because I am in a lot of trouble.

–San Diego Reader

EXAMPLE: DEAR ABBY…

According to well-documented norms, distribution of gestation time is approximately normal with mean 266 days and SD 16 days.

What percent of babies have a gestation time greater than or equal to 310 days (10 months and 5 days)? ↳ Right tail

4¥ = 31016T = 2.75

#hE⇐ '

p ft£2.75 ) s. 997

o :P ft> 2.75 ) = 1- . gay= o .

003

NORMAL CALCULATIONS

When presented with a problem that involves calculations of normal probabilities:

(a) Always draw a picture

(b) Standardize

(c) Recall: Once we have standardized, only need a

single table of probabilities for N(0,1). Why?

EYE's

0

EXAMPLE: DEAR ABBY…

EXCEEDING AIR QUALITY STANDARDS

Javits studied the number of daily exceedances of air quality standards per year in the U.S. A daily exceedance occurs if the ambient pollutant concentration is exceeded in a 24-hour period. Assume that each of the 365 days (exclude leap years) has a 10% chance of exceeding the ambient pollutant concentration.

(a) What is the distribution of the total number of days in a year where the pollution concentration is exceeded?

(b) How many days do you expect that the pollution concentration will exceed the air quality standards?

(c) What is the probability that the number of daily exceedances is at least 30 days?

Bin ( 365 ;D×= total ¥0445 .

oI36s×E(X)=nP = 365×-1=26.5

-lt.Fa

The Variance of < Bono much , p)OZ n p ( tp )

= 365 × o . l × ( t - o . Do = ± FEE

pTXzftp(×3o )tPdEp(×z3o)

PH23D= p H⇒Dtp ( x =3 ) )+ p(x⇒Dt .

. + PH⇒ 4)

01-0.108=14×230)=

0.892

EXAMPLE: GUESS A NUMBER

During a psychology experiment a subject is asked to randomly pick a number between 0 and 1. Assume that each number has an equal probability of being chosen. Find the probability of the following events:

1. The subject picks a number between 0.3 and 0.7.

2. The subject picks a number less than 0.5 or greater than 0.8.

. 45

•7=prub .

UNIFORM DISTRIBUTION

I

KKHXha , a

=. 4

Continuous disfn ,

1--3±4

P(X=-3=0

Etsu+ .

2 =. 7

pick a random number

between 0,2

' ¥*2×1=1

.