DISCRETE RANDOM VARIABLES Discrete random variable A discrete
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, . . . 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.
Relative frequency Number of flat sides up. 0 1 2 3 4 Score 5 1 2 3 4 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
Listing possibilities: have a system 3 sticks 2 sticks 1 stick f f c c c f f f c c c f f c c c f f f f c c f f f c f c c f f f c c f c f c c f f c c c c f c c
f f f f c c f f 4 x P(f, f, f, c)= 0. 73 x 0. 3 x 4= f c 6 x P(f, f, c, c)=0. 72 x 0. 32 x 6 = f c c f f c c c 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= 0. 2401 = 0. 74 4 x P(f, c, c, c) = 0. 7 x 0. 33 x 4 = 3 sticks 1 x P(c, c, c, c) = 0. 34 = f f f c c f f c f c f f c c c f c c 2 sticks 1 stick f c f f f c c c f f c f c c c c c
Relative frequency Number of flat sides up. 0 1 2 3 4 Score 5 1 2 3 4 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 s. 1 2 3 4 5 P(S=s) Probability function 0. 0756 0. 2646 0. 4116 0. 2401 0. 0081 Note s is used for individual values of the random variable S
P(X=x) as a stick/bar graph
TASK Exercise A – Page 53 & 54 Questions: 1, 2, 3, 5 & 6 Do rest at home.
Mean, variance and standard deviation s. 1 P(S=s) Probability function 0. 0756 0. 2646 2 3 4 0. 4116 0. 2401 5 0. 0081 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
Mean, variance and standard deviation s. 1 P(S=s) Probability function 0. 0756 0. 2646 2 3 4 0. 4116 0. 2401 5 0. 0081 However, multiplying and dividing by 10000 both top and bottom seems unnecessary and it is
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 B 1, B 2
VARIANCE x 0 1 2 3 4 P(X=x) Probability function 0. 15 0. 25 0. 1 Mean = 0 x 0. 15 + 1 x 0. 25 + 2 x 0. 25 + 3 x 0. 25 + 4 x 0. 1=1. 9 x 0 x-μ -1. 9 (x-μ)2 P(X=x) (x-μ)2 x P(X=x) 3. 61 0. 15 0. 5415 1 2 3 4 -0. 9 0. 1 1. 1 2. 1 0. 81 0. 01 1. 21 4. 41 0. 25 0. 1 0. 2025 0. 0025 0. 3025 0. 4410
Variance & Standard deviation x 0 1 2 3 4 x-μ -1. 9 (x-μ)2 P(X=x) (x-μ)2 x P(X=x) 3. 61 0. 15 0. 5415 -0. 9 0. 81 0. 25 0. 1 0. 01 0. 25 1. 1 1. 21 0. 25 2. 1 4. 41 0. 1 Variance = σ2 Standard deviation = σ 0. 2025 0. 0025 0. 3025 0. 4410 1. 49 1. 22 The standard deviation or random variables is normally denoted as σ
TASK Page 56 question 2 Homework – test yourself
- Slides: 14