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# Statistics lecture 6 (ch5)

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Probability Distributions

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### Statistics lecture 6 (ch5)

1. 1. Probability distributions • Discrete – Binomial distribution – Poisson distribution • Continuous – Normal distribution
2. 2. • Discrete random variable – Variable is the characteristic of interest that assumes different values for different elements of the sample/population. – If the value of the variable depends on the outcome of an experiment it is called a random variable. – Discrete random variable takes on a countable number of values.
3. 3. • Discrete distribution function - example – Toss a coin twice. – S = {HH; HT; TH; TT} – Each outcome in S has a probability of ¼. – Random variable X – number of heads – Collection of probabilities – probability distribution – associates a probability with each value of random variable. x 0 1 2 1 2 1 P(X = x) = P(x) 4 4 4
4. 4. • Discrete distribution function – 0 ≤ P(x) ≤ 1, for each x – ∑P(x) = 1 x 0 1 2 1 2 1 P(X = x) + + =1 4 4 4 1 3 4 P(X ≤ x) 4 4 4
5. 5. Let X denote the number of defective memory chips that arereturned to the production plant in a production batch of 300.Thenumber of returns received varies from 0 – 4. x 0 1 2 3 4 P(x) 0.15 0.3 0.25 0.2 0.1Use the probability distribution given above to calculate:-1. The probability that exactly 3 memory chips are returned2. The probability that more than two memory chips are returned3. That at least two memory chips are returned4. From 1-3 memory chips are returned5. Less than 2 memory chips are returned6. At the most 2 memory chips are returned7. Between one and four memory chips are returned
6. 6. Answers • Example 5.3, p154 Elementary Statistics
7. 7. MEAN • Represents average value that we expect to obtain if the experiment is performed a large number of times   E ( X )   xP( x)STANDARD DEVIATION• SD gives a measure of how dispersed around the mean the variable is    x P( x)   2 2
8. 8. • Discrete distribution function – Mean =   E ( X )   xP( x) – expected value – St dev =    x 2 P( x)   2 0 1 2 1 2 1 P(X = x) 4 4 4 1 2 1   xP( x)  0    1   2    1 4 4 4  x 2 P( x)   2  02  1 4  12  2 4  22  1 4  12  0.71
9. 9. • Discrete distribution function - example – A survey was done to determine the number of vehicles in a household. – A sample of 560 households was taken and the number of cars was captured. – Random variable X – number of cars. – The results are: x – Number of cars 0 1 2 3 4 Number of 28 168 252 79 33 households
10. 10. • Discrete distribution function - example x – Number of cars 0 1 2 3 4 Number of 28 168 252 79 33 560 households P(X = x) 0.05 0.30 0.45 0.14 0.06 252 28 168 P( X  1)  0)  2)  0.05  0.30 0.45 560 560
11. 11. • Discrete distribution function - example x – Number of cars 0 1 2 3 4 Number of 28 168 252 79 33 households P(X = x) 0.05 0.30 0.45 0.14 0.06 1
12. 12. • Discrete distribution function - example x – Number of cars 0 1 2 3 4 P(X = x) 0.05 0.30 0.45 0.14 0.06   xP( x) 0  0.05  1 0.30   2  0.45   3  0.14   4  0.06   1.86  x 2 P( x)   2 0  0.05   1  0.3  2  0.45   3  0.14   4  0.06   1.86 2 2 2 2 2 2 0.93
13. 13. • Discrete distribution functions – Binomial distribution – Poisson distribution P( X  x)  p( x)  n C x p x (1- p ) n - x
14. 14. • Continuous random variable – Random variable that takes on any numerical value within an interval. – Possible values of a continuous random variable are infinite and uncountable. – Obtained by measurement has a unit of measurement associated to it.
15. 15. •• Ascontinuous random variable has probability of each A the number of outcomes increases the an uncountable value decreases. infinite number of values in the interval (a,b) • This is so because the sum of all the probabilities remains 1. • The probability that a continuous variable X will assume any particular value is zero. Why? • When the number of values approaches infinity (because X is continuous) the probability of each value approaches 0. The probability of each outcome4 outcomes 1/4 + 1/4 + 1/4 + 1/4 =13 outcomes 1/3 + 1/3 + 1/3 =12 outcomes 1/2 + 1/2 =1
16. 16. • A lot of continuous measurement will become a smooth curve.• The probability density curve describe the probability distribution. Area = 1• The density function satisfies the following conditions: – The total area under the curve equals 1. – The probability of a continuous random variable can be identified as the area under the curve.
17. 17. • The probability that x falls between a and b is found by calculating the area under the graph of f(x) between a and b. P(a < X < b) a b
18. 18. • Continuous distribution functions – Normal distribution