Statistical Experiment A statistical experiment or observation is any process by which an...
Transcript of Statistical Experiment A statistical experiment or observation is any process by which an...
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Statistical Experiment
A statistical experiment or observation is any process by which an measurements are
obtained
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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
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Random Variable
a quantitative variable that assumes a value determined by
chance
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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
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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
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Probability Distribution
an assignment of probabilities to the specific values of the random variable or to a range of values of
the random variable
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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.
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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
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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
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1
Probabilities must be between zero and one (inclusive)
P(x) =116
6
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Probability Histogram
P(x)
1 2 3 4 5 6
| | | | | | |6
1
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Mean and standard deviation of a discrete probability
distribution
Mean = = expectation or expected value,
the long-run average
Formula: = x P(x)
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Standard Deviation
)x(P)x( 2
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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
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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
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Standard Deviation
1.28
64.1)x(P)x( 2
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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.
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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
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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 =
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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.
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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
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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