sections 8.5 & 8.6 & 5.1 & 5.2 & 5 - math.tamu.edu

Sections 8.5 & 8.6 & 5.1 & 5.2 & 5.3

Tekin Karadag

Texas A&M University

Department of MathematicsTexas A&M University, College Station

29-31 July, 2020

Tekin Karadag (TAMU) Finite Mathematics Week4 1 / 41

1 The Normal Distribution

2 Applications of the Normal Distribution

3 Compound Interest

4 Annuities

5 Amortization and Sinking Funds


The Normal Distribution

In the previous sections, we focused especially on discrete random variablesand represented them in a histogram. In this section, our aim is tounderstand continuous random variables. NOTE: The random variableassociated with the standard normal distribution is designated by Z .


Definition

The random variable X has a normal distribution on the interval(−∞,∞) if the probability P(a ≤ X ≤ b) that X is between a and b isthe area under the standard normal curve given by

y =1

σ√

2πexp−0.5[ (x−µ)

σ]2

on the interval [a, b], where π ≈ 3.14159 and exp ≈ 2.71828 and σ is thestandard deviation and µ is the mean (expected value).


Remark

Characteristic of the Normal Curve The graph of the normal curve hasthe following characteristics.

1) It is bell shaped.

2) It is symmetric about x = µ.

3) It lies above the x-axis.

4) It approaches but is never equal to 0 on both the positive andnegative x-axis.

5) The area under the entire curve is exactly 1.


Remark

TI 83/84 calculators have three functions that are used for normalprobability calculations. Access the Distributions menu by pressing2nd and then VARS. The 1:normalpdf( command will find the normalprobability density function for a given value of Z .

The 2:normalcdf( command finds area under the normal probabilitydensity function. The first value entered is left endpoint and thesecond value entered is the right endpoint.

If, for example, a normal probability distribution has a mean of 100and a standard deviation of 15, we find P(70 ≤ X ≤ 80) asnormalcdf(70,80,100,15). In general, it is given asnormalcdf(leftpoint,rightpoint,mean,stand.dev).

Suppose you want to find c such that P(X ≤ c) = 0.75 on a normalprobability distribution with a mean µ of 20 and standard deviation σof 5. Then again go to Distributions menu select 3:invNorm(. EnterinvNorm(0.75,20,5) to find c . In general, it is given asinvNorm(probability,mean,stand.dev).


Definition

When σ = 1 and µ = 0, we call the distribution as standard normaldistribution.

Example

Let Z be a random variable with standard normal distribution (µ = 0 andσ = 1)

a) Find P(0.24 < Z < 1.48).

b) Find P(Z < 1.24).

c) Find P(Z > 0.5).

d) Find c when P(Z ≤ c) = 0.72 .

Solution:


Example

Let X be a random variable. Find the following probabilities when µ and σgiven.

a) Find P(X ≤ 50) when µ = 38, σ = 8.

b) Find c when P(Z ≤ c) = 0.97 when µ = 100, σ = 50.

c) Find P(X ≥ 0.01) when µ = 0.006, σ = 0.002.

d) Find P(10 ≤ X ≤ 20) when µ = 5, σ = 10.

Solution:


Example

(a) Let Z be the standard normal variable. Find a such that

P(−a < Z < a) = 0.4131.

(b) Let Z be the normal variable with mean µ = 60 and standarddeviation σ = 3.5. Find A and B such that

P(A < Z < B) = 0.7923

if A and B are symmetric about the mean.

Solution:


Applications of the Normal Distribution

Example

The weight, in pounds, of a certain type of adult squirrel is normallydistributed with a mean of 3 pounds and a standard deviation of 0.50pound. What percentage of these squirrels have weight

(a) less than 2 pounds

(b) greater than 4 pounds

(c) between 2 and 4 pounds

Solution:


Example

The grade point average (GPA) of the senior class of Jefferson HighSchool is normally distributed with a mean of 2.7 and a standard deviationof 0.4. If a senior in the top 10% of his or her class is eligible for admissionto any of the nine campuses of the state university system, what is theminimum GPA that a senior should have to ensure the eligibility foruniversity admission?

Solution:


The Binomial Distribution revisited

Remark

The mean of the binomial distribution with n trials and the probabilityof ”success” equal to p and of ”failure” equal to q is

µ = np,

the variance Var(X ) isVar(X ) = npq

and the standard deviation is

σ(X ) =√

npq.


Example

A new drug has been found to be effective in treating 75 % of the peopleafflicted by a certain disease. If the drug is administered to 800 peoplewho have this disease, what are the mean(expected value) and thestandard deviation of the number of people for whom the drug can beexpected to be effective?

Solution:


SIMPLE INTEREST

We previously learned compound interest but it is not the only interesttype. We will see one other type of interest, simple interest, and learncompound interest deeper this week.

Definition

Suppose a sum of money P, called the principal or present value, isinvested for t years at an annual simple interest rate of r , where r isgiven as a DECIMAL. Then the interest I for one year is

I = Pr ,

and for t years isI = Prt.

The future value or accumulated amount A at the end of t years is

A = P + I = P + Prt = P(1 + rt).


Example

Find how much interest and accumulated amount were earned at the endof 6 months on a $2000 bank deposit paying simple interest of 8% a year.

Solution:

Example

A bank deposit paying simple interest at 7% a year grew to a sum of$1400 in 15 months. What was the principal amount?

Solution:


PERIODICAL COMPOUND INTEREST

Compound interest is more common type of interest that almost all of thebanks use. As we saw previously, the compound interest is periodicallyadded to the principal, and that money will earn interest at the same rate.There are two types of compound interest:

Definition

Periodically Compounded Interest: Suppose a principal P earns at theannual rate of r , expressed as a decimal, and interest is compounded mtimes a year. Then the amount A after t years is

A = P(

1 +r

m

)mt,

where mt represents the number of time periods andr

mrepresents the

interest per period.


Example

Find the present value of $45,000 due in 4 years at the rate of 7% interestper year compounded monthly. (Round your answer to the nearest cent.)

Solution:


Example

Nine years ago, Sam invested $15,000 in a retirement fund that grew atthe rate of 10.78% per year compounded annually. What is his accountworth today? (Round your answer to the nearest cent.)

Solution:


CONTINUOUS COMPOUND INTEREST

Recall that we discussed continuous compound interest as an applicationof exponential functions in Section 5.3.

Definition

Continuously Compound Interest:Suppose a principal P earns at theannual rate of r , expressed as a decimal, and interest is compounded eachinstant. Then the amount A after t years is

A = Pert .

Example

A bank offers an account that earns 4.16% interest per year compoundedcontinuously. If a person invests $8,000 into the account, what will be thevalue of the account at the end of 6 years? (Round to the nearest cent.)

Solution:


EFFECTIVE RATE OF INTEREST

For a bank account, we see that simple interest rate is good, periodical compoundinterest is better, and continuous compound interest is the best. It is opposite forloans, we want lower rates and compounded less. So, how can we comparedifferent accounts or loans?

DefinitionSuppose P is the principal amount invested at an annual rate of r expressed as adecimal and is compounded m times a year. And suppose A is the accumulatedamount after a year. Then, the effective annual yield is the annual rate of reff

that gives the same accumulated amount A at the end of a year withoutcompounding.

Example:


Remark

For periodically compounded accounts:

reff =(

1 +r

m

)m− 1,

For continuously compounded accounts:

reff = er − 1.


Example

Find the effective rate of interest corresponding to annual rate of 9.5%compounded in the following ways. (Round answers to two decimalplaces.)

1 compounded annually

2 compounded semiannually

3 compounded quarterly

Solution:


Example

Mr. Jackson is trying to decide on a bank account. Account A iscompounded every 4 months at an annual rate of 5.6%. Account B iscompounded continuously at an annual rate of 5.4%. Which account ismaking more money?

Solution:


FINDING reff BY USING CALCULATOR

Here are the steps for TI-83 plus or TI-84:

1 Press APPS

2 Press 1

3 Select C:IEff(

4 Fill as following: Eff(annual interest rate as a percentage(NOT indecimal), the number of compounding periods per year)


Example

A family wants to get a loan lasting 30 years for $250,000 to buy a house.Loan company A offers a rate of 8.2% per year compounded every 2months. Loan B is compounded semiannually (twice a year) at a rate of8.4% per year. Use effective yield rates to determine which loan is betterfor the family.

Solution:


Present Value

Definition

Suppose an account earns an annual rate of r expressed as a decimal andcompounds m times a year. Then the amount P, called the presentvalue, needed currently in this account so that a future amount of F willbe obtained in t years is given by

P =F(

1 + rm

)mt .


Example

What is the present value if you compounded the principal (or presentvalue) daily with an interest rate 5.4 and get $7900 at the end of 2 years?

Solution:


Example

How much money must grandparents set aside to a bank offering aninterest rate 9% which is compounded quarterly at the birth of theirgrandchild if they wish to have $20,000 when the grandchild reaches his orher 18th birthday?

Solution:


Doubling Times

Example

Find the time for a $1000 investment compounding annually at an annualrate of 6% to double.

Solution:


Example

Find the time for an account earning interest compounded annually at therate of 9% to grow from $2000 to $5000.

Solution:


Annuities

DefinitionAn annuity is a sequence of equal payments made at equal time periods.

Example

An individual is trying to save money for a down payment on a house purchasedin five years. She can deposit $100 at the end of each month into an account thatpays interest at an annual rate of 9% compounded monthly. How much is in thisaccount after five years? Also, find the amount of interest that has been earned.

Solution:


Example

Every six months an individual places $1000 into an account earning anannual rate of %10 of compounded semiannually. Find the amount in theaccount at the end of 15 years.

Solution:


Present Value of Annuities

In order to find the present value of annuities, we need to find PV in TVM Solver.

Example

You have announced to your company that you will retire in one year. Yourpension plan requires the company to pay you $ 25000 in a lump sum at the endof one year and every year thereafter until your demise. The company makes theassumption that you will live to receive 15 payments. Interest rates are 7% peryear compounded annually. What amount of money should the company set asidenow to ensure that they can meet their pension obligations to you?

Solution:


Example

Two oil wells are for sale. The first will yield payments of $12,000 at theend of each of the next 11 years, while the second will yield $6,000 at theend of each of the next 23 years. Interest rates are 4.5% per year over thenext 23 years. Which has the higher present value?

Solution:


Amortization and Sinking Funds

AmortizationFor finding the amount needed to amortize a loan or a borrowed money,we will find PMT, the periodic payments, which are needed to pay a debtin n periods with an interest rate i per period.

Example

You wish to borrow $12000 from the bank to purchase a car. The bankcharges interest at an annual rate of 12%. There are to be 48 equalmonthly payments with the first to begin in one month. What must thepayments be so that the loan will be paid off after 48 months ?

Solution:


Example

Find the monthly payment needed to amortize a typical $180,000mortgage loan amortized over 30 years at an annual interest rate of 4.5%compounded monthly. Find the total interest paid on the loan.

Solution:


Amortization Schedule

We now look at what happens if you wish to make a lump-sum payment to payoff a loan. For example, it is routine for an individual to sell a house and buyanother. This requires paying off the old mortgage. How much needs to be paid?To answer this, we look at closely at how much is owed at the end of eachpayment period. The next example indicates how to create such a schedule.

Example

You agree to sell a small piece of property and grant a loan of $9000 to the buyerwith annual interest at 10% compounded annually, with payments of equalamounts made at the end of each of the next six years. Construct a table thatgives for each period the interest, payment toward principal, and the outstandingbalance.

Solution:


Finding the Outstanding Balance

Example

James secures a bank loan of $ 300,000 to purchase a house. The term ofthe mortgage is 30 years, and the interest rate is 8% per year compoundedmonthly on the unpaid balance.

(a) What is James’ current monthly mortgage payment?

(b) What is James’ current outstanding balance after 60 payments?

Solution:


An Equity Problem

Example

A family has purchased a house for $130,000. They made an initial downpayment of $10,000 and secured a mortgage with interest charged at the rate of9% year compounded monthly on the unpaid balance. The loan is to beamortized over 30 years.

(a) What monthly payment will the family be required to make?

(b) How much total interest will they pay on the loan?

(c) What will be their equity after 10 years?

(d) What will be their equity after 22 years?

Solution:


Sinking Funds

Often an individual or corporation knows at some future date a certain amountFV of money will be needed. Any account that is established for accumulatingfunds to meet a future need is called a sinking fund. In this case, FV , n and iare known and one wishes to calculate the periodic payment PMT .

Example

A corporation wishes to set up a sinking fund in order to have the funds necessaryto replace a current machine. It is estimated that the machine will need to bereplaced in 10 years and cost $100,000. How much per quarter should bedeposited into an account with an annual rate of 8% compounded quarterly tomeet this future obligation? What will be the total amount of the payments andwhat will be the interest earned ?

Solution:


Example

A corporation creates a sinking fund in order to have $540,000 to replace somemachinery in 11 years.

(a) How much should be placed in this account at the end of each quarter ifthe annual interest rate is 6.3% compounded quarterly?

(b) How much interest would they earn over the life of the account?

(c) Determine the value of the fund after 6 years.

(d) How much interest was earned during the third quarter of the 4th year?

Solution:


sections 8.5 & 8.6 & 5.1 & 5.2 & 5 - math.tamu.edu

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