Statistical Experiment

A statistical experiment or observation is any process by which an measurements are

obtained

Examples of Statistical Experiments

• Counting the number of books in the College Library

• Counting the number of mistakes on a page of text

• Measuring the amount of rainfall in your state during the month of June

Random Variable

a quantitative variable that assumes a value determined by

chance

Discrete Random Variable

A discrete random variable is a quantitative random variable that can take

on only a finite number of values or a countable number of values.

Example: the number of books in the College Library

Continuous Random Variable

A continuous random variable is a quantitative random variable that can take on any of the countless number of

values in a line interval.

Example: the amount of rainfall in your state during the month of June

Probability Distribution

an assignment of probabilities to the specific values of the random variable or to a range of values of

the random variable

Probability Distribution of a Discrete Random Variable

• A probability is assigned to each value of the random variable.

• The sum of these probabilities must be 1.

Probability distribution for the rolling of an ordinary die

x P(x)

1

2

3

4

5

6

6

1

6

1

6

1

6

1

6

1

6

1

Features of a Probability Distribution

Features of a Probability Distribution

x P(x)

1

2

3

4

5

6

6

1

6

1

6

1

6

1

6

1

6

1

Probabilities must be between zero and one (inclusive)

P(x) =116

6

Probability Histogram

P(x)

1 2 3 4 5 6

| | | | | | |6

1

Mean and standard deviation of a discrete probability

distribution

Mean = = expectation or expected value,

the long-run average

Formula: = x P(x)

Standard Deviation

)x(P)x( 2

Finding the mean:

x P(x) x P(x)

0 .3

1 .3

2 .2

3 .1

4 .1

0

.3

.4

.3

.4

1.4

= x P(x) = 1.4

Finding the standard deviationx P(x) x – ( x – ) 2 ( x – ) 2 P(x)

0 .3

1 .3

2 .2

3 .1

4 .1

– 1.4

– 0.4

.6

1.6

2.6

1.96

0.16

0.36

2.56

6.76

.588

.048

.072

.256

.676

1.64

Standard Deviation

1.28

64.1)x(P)x( 2

Linear Functions of a Random Variable

If a and b are any constants and x is a random variable, then the new random

variable

L = a + bx

is called a linear function of a random variable.

If x is a random variable with mean and standard deviation

, and L = a + bx then:

• Mean of L = L = a + b

• Variance of L = L 2 = b2 2

• Standard deviation of L = L= the square root of b2 2 = b

If x is a random variable with mean = 12 and standard

deviation = 3 and L = 2 + 5x• Find the mean of L.

• Find the variance of L.

• Find the standard deviation of L.

L = 2 + 5

• Variance of L = b2 2 =

25(9) = 225

• Standard deviation of L = square root of 225 =

Independent Random Variables

Two random variables x1 and x2

are independent if any event involving x1 by itself is

independent of any event involving x2 by itself.

If x1 and x2 are a random variables with means and and

variances and

If W = ax1 + bx2 then:

• Mean of W = W = a + b

• Variance of W = W 2 = a2 12 + b2 2

• Standard deviation of W = W= the square root of a2 1

2 + b2 2

Given x1, a random variable with 1 = 12 and 1 = 3 and x2 is a

random variable with 2 = 8 and 2

= 2 and W = 2x1 + 5x2. • Find the mean of W.

• Find the variance of W.

• Find the standard deviation of W.

• Mean of W =

2(12)+ 5(8) = 64• Variance of W =

4(9) + 25(4) = 136

• Standard deviation of W= square root of 136 11.66