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DISCRETE & RANDOM VARIABLES

DISCRETE & RANDOM VARIABLES. Discrete random variable. A discrete random variable is one which may take on only a countable number of distinct values such as 0, 1, 2, 3, 4, ...

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DISCRETE & RANDOM VARIABLES

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  1. DISCRETE & RANDOM VARIABLES

  2. Discrete random variable • A discrete random variable is one which may take on only a countable number of distinct values such as 0, 1, 2, 3, 4, ... • Discrete random variables are usually (but not necessarily) counts. If a random variable can take only a finite number of distinct values, then it must be discrete. • Examples of discrete random variables include the number of children in a family, the Friday night attendance at a cinema, the number of patients in a doctor's surgery, the number of defective light bulbs in a box of ten.

  3. Relative frequency • The only way to get an estimate of the probability is to throw the sticks many times and find what we call the: • relative frequency • Q Can be different each time why? • If the stick is thrown 100 times and the results are: • Flat side up: 70 giving an estimate probability of 0.7 • Curved side up: 30 giving an estimate probability of 0.3

  4. Listing possibilities:have a system 3 sticks 2 sticks 1 stick

  5. Working out probabilities: Each stick falls independently, f=0.7 c=0.3 1 x P(f,f,f,f) = 0.7 x 0.7 x 0.7 x 0.7= 0.2401 = 0.74 4 x P(f,f,f,c)= 0.73 x 0.3 x 4= 6 x P(f,f,c,c)=0.72 x 0.32 x 6 = 4 x P(f,c,c,c) = 0.7 x 0.33 x 4 = 1 x P(c,c,c,c) = 0.34 = 3 sticks 2 sticks 1 stick

  6. Relative frequency • The score is an example of discrete random variable. • Let S stand for score. Capital letters are used for random variables. • P(S=3) means ‘the probability that S=3 • P(S=3) = 0.4116 Note s is used for individual values of the random variable S

  7. P(X=x) as a stick/bar graph

  8. TASK • Exercise A – Page 53 & 54 • Questions: 1, 2, 3, 5 & 6 • Do rest at home.

  9. Mean, variance and standard deviation If I were to throw 10000 times, I could work out the mean like the below. Multiply each of my probabilities by 10000 and then divide by 10000

  10. Mean, variance and standard deviation However, multiplying and dividing by 10000 both top and bottom seems unnecessary and it is

  11. MEAN of: Discrete random Variables • Mean S =Σs x P(S=s) • The mean of a random variable is usually denoted by μ (‘mu’) Task B1, B2

  12. VARIANCE • Mean = 0x0.15 + 1x0.25 + 2x0.25 + 3x0.25 + 4x0.1=1.9

  13. Variance & Standard deviation The standard deviation or random variables is normally denoted as σ

  14. TASK • Page 56 question 2 • Homework – test yourself

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