MTK3005 Chapter 2

8/8/2019 MTK3005 Chapter 2

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MTK3005 Engineering Statistics

Chapter 2 Random Variables and ProbabilityDistributions

Chapter Outline

2.1 Random Variables 2.2 Discrete Probability Distributions

2.3 Binomial Distributions 2.4 Poisson Distributions

2.1 Random variables

A random variable is a numerical variable whose measured value can change from one replicate of theexperiment to another.

A n uppercase letter is used to denote a random variable.A lowercase letter is used to denote the measured value of the random variable.

There are two types of random variables:i. Discrete random variables

-random variable with a finite number of values or a countable number of values.-Example: number of scratches on a surface, proportion of defective product among1000 tested

ii. Continuous random variables-random variable with an interval of real numbers for its range.-Example: electrical current, pressure, temperature, time

Ex ample 2.1 A balanced coin is tossed three times. List the elements of the sample space and corresponding values of the random variable , the total number of heads. If H for heads and T for tails

Solution:

Event HHH 3HHT 2HTH 2THH 2HTT 1THT 1TTH 1TTT 0

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2.2 Discrete Probability Distributions

The set of ordered pairs is a probability function, probability mass function or probability

distribution of the discrete random variable if, for each possible outcome ,

1

2. 1)( !§ x P

Ex ample 2.2 From example 2.1, find the probability distribution for X.

Event ProbabilityHHH 3

HHT 2

HTH 2 THH 2

HTT 1

THT 1

TTH 1

TTT 0

0 1 2 3

0.375 0.125

Ex ample 2.3 Below is the number of goal that was scored by 40 football teams in a football league. Find the

probability distribution. X represents the number of goal.

No. of goal Frequency, 0 41 62 123 10

4 8S olution

0 1 2 3 4

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Ex ample 2.4G iven the probability distribution as below.Find (i) (ii) (iii)

Solution(i) 25.03 !! x

(ii) 2102 !!!!e x¡

x¡

x¡

x¡

= 3.015.01.0 = 55.0

(iii) 4 5.02.025.04342 !!!!!e x¢

x¢

x¢

2.2.1 Mean and variance

The mean or expected value of the discrete random variable is

The variance of is

Ex ample 2.5

Determine the mean and variance.0 1 2 3 4

S olutionM ean, 3.22.0425.033.0215.011.00 !!! x E Q

0 -2.3 5.29 0.1 0.5291 -1.3 1.69 0.15 0.25352 -0.3 0.09 0.3 0.0273 0.7 0.49 0.25 0.12254 1.7 2.89 0.2 0.578

= 1.51

2.3 Binomial Distribution

A n experiment often consists of repeated trials, each with two possible outcomes that may be labeledsuccess or failure. The process is referred to as Bernoulli process.The Bernoulli process must possess the following properties:

1. The experiment consists of repeated trials.2. Each trial results in an outcome that may be classified as a success or a failure.3. The probability of success, denoted by , remains constant from trial to trial.4. The repeated trials are independent.

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A Bernoulli trial can result in a success with probability and a failure with probability . Thenthe probability distribution of the binomial random variable , the number of success in independenttrials, is

Ex ample 2.6The probability that a certain kind of component will survive a given shock test is . Find theprobability that exactly 2 of the next 4 components tested survive.

S olution

For each of the tests,

The mean and variance of the binomial distribution are

and

2.4 Poisson Distribution

The probability distribution of the Poisson random variable , representing the number of outcomesoccurring in a given time interval or specified region denoted by , is

!

: x

e x p

x Q

Q

Q

!

Ex ample 2.7During a laboratory experiment the average number of radioactive particles passing through a counterin 1 millisecond is 4. What is the probability that 6 particles enter the counter in a given millisecond?

S olution

and

Ex ample 2.8Ten is the average number of ship arriving each day at a certain port. The facilities at the port can handleat most 15 ships per day. What is the probability that on a given day ships have to be turned away?

S olution

The mean and variance of the Poisson distribution both have the value .

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MTK3005 Chapter 2