How to solve probability distributions?, Study notes of Business Mathematics

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Discrete Probability Distributions

Random Variable

 (^) Random experiment is an experiment with

random outcome.

 (^) Random variable is a variable related to a

random event

Discrete Random Variables

 (^) The number of throws of a coin needed before a

head first appears

 (^) The number of dots when rolling a dice

 (^) The number of defective items in a sample of 20

items

 (^) The number of customers arriving at a check-out

counter in an hour

 (^) The number of people in favor of nuclear power in

a survey

Continuous Random Variables

 (^) The yearly income for a family

 (^) The amount of oil imported into Finland in a

particular month

 (^) The time that elapses between the installation of a

new component and its failure

 (^) The percentage of impurity in a batch of chemicals

Fortune wheel

If the probability to win when rolling a fortune wheel is 15% then the probability distribution for the number of wins in 5 rolls is:

number of wins probability 0 44,3705% 1 39, 2 13,8178% 3 2,4384% 4 0,2152% 5 0,0076%

Two Dice

0

1/

2/

3/

4/

5/

6/

7/

0 1 2 3 4 5 6 7 8 9 10 11 12 13 Sum of outcomes

Probability

Expected Value

 (^) Expected value is just like the mean in empirical distributions

Examples:  (^) When playing a dice the expected value equals 3,  (^) Insurance company is interested in the expected value of indemnities  (^) Investor is interested in the expected value of portfolio’s revenue

Expected value calculation

 (^) The expected value for a discrete random variable

is obtained by multiplying each possible outcome by its probability and then sum these products

Expected Value example 2

 (^) Assume a lottery with 1000 lottery and 31 winning

tickets. One ticket wins 500, ten tickets win 300 and 20 tickets win 100.

 (^) Define the ticket price so that the expected value of

the win is 55% of the ticket price.

Expected value example 3

According to manufacturer’s statistics the car model needs repairs under warranty as follows: No repairs for 50% of cars

On the average 100 euros repairs for 20% of cars

On the average 200 euros repairs for 25% of cars

On the average 500 euros repairs for the rest of the

cars

How much should the warranty increase the price of the car?

Binomial Distribution Bin(n,p)

The experiment consists of a sequence of n identical trials

All possible outcomes can be classified into two

categories, usually called success and failure The probability of an success, p, is constant from trial to

trial The outcome of any trial is independent of the outcome of

any other trial

Binomial experiments satisfy the following:

Binomial Distribution Random Variables

 (^) The number of heads when tossing a coin for 50

times

 (^) The number of reds when spinning the roulette

wheel for 15 times

 (^) The number of defective items in a sample of 20

items from a large shipment

 (^) The number of people in favour of nuclear power in

a survey

Poisson distribution

Poisson experiments satisfy the following

 (^) The probability of occurrence of an event is the

same for any two intervals of equal length

 (^) The occurrence or non-occurrence of the event in

any interval is independent of the occurrence or non-occurrence in any other interval

 (^) The probability that two or more events will occur in

an interval approaches zero as the interval becomes smaller

Poisson Distribution Random Variables

 (^) The number of failures in a large computer system

during a given day  (^) The number of ships arriving at a loading facility

during a six-hour loading period  (^) The number of delivery trucks to arrive at a central

warehouse in an hour  (^) The number of dents, scratches, or other defects in

a large roll of sheet metal  (^) The number of accidents at a crossroads during

one year