04 random-variables-probability-distributionsrv

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  • 1. Quantitative Methods Varsha Varde
  • 2. PROBABILITY DISTRIBUTION We Studied frequency distribution of raw data This helps us in knowing which outcomes occurred most often, which occurred least often etc. After studying a large number of such patterns of data, statisticians derived a number of model patterns or model distributions These distributions are called probability distributions They combine the concept of frequency distribution with the concept of probability Probability distributions are of two types: Discrete and Continuous Varde Varde 2
  • 3. Probability Distribution A probability distribution is an array (or arrangement) that shows the probabilities of individual possible outcomes or of different values of a variable. The sum of all the probabilities in a probability distribution is always equal to one, that is, if X is a variable with N possible values X1, X2, , XN taken with probabilities P (X1), P (X2), .., P (XN) then N P (Xi) = 1 i = 1 Varde Varde 3
  • 4. Discrete Probability Distribution A probability distribution is said to be discrete if the values of the corresponding random variable are discrete. We shall describe two typical discrete probability distributions namely, Binomial and Poisson distributions Varde Varde 4
  • 5. Continuous Probability Distribution A probability distribution is continuous if the values of the corresponding random variable are continuous, that is, they fall in an interval. We shall study one typical continuous distribution, namely, normal distribution . Three more typical distributions of continuous type ( t distribution, chi square distribution and F distribution) Varde Varde 5
  • 6. Mean of a Probability Distribution The mean of a probability distribution is the number obtained by multiplying all the possible values of the variables by the respective probabilities and adding these products together. It indicates the expected value the corresponding variable would take. It is generally denoted by the greek letter or E(x) Varde Varde 6
  • 7. Variance of Probability Distribution Variance of a probability distribution is the number obtained by multiplying each of the squared deviations from the mean by its respective probability and adding these products. It is generally denoted by the greek letter 2 or v(x) Varde Varde 7
  • 8. Formulae for Mean & Variance If X is a variable with N possible values X1, , XN taken with probabilities P (X1), P (X2), P (XN) then the expected value or the mean of this probability distribution is N E(x)= = Xi P (Xi) i = 1 variance 2 of this probability distribution is N 2 = (Xi )2 P (Xi) i = 1 Standard Deviation = V (X) = (x - )2 p(x)Varde Varde 8
  • 9. EXAMPLE Roll a pair of dice and work out the probability distribution of the sums of the spots that appear on the faces. The variable sum of dots on two faces would lie between 2 and 12 The probability distribution of this variable would be: Sum: 2 , 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 Prob: 1/36, 2/36 , 3/36, 4/36, 5/36 ,6/36 ,5/36, 4/36, 3/36, 2/36, 1/36 It can be seen that the sum of the probabilities in this case is 36 / 36 = 1 indicating that this is a complete probability distribution. Varde Varde 9
  • 10. EXAMPLE The mean of this probability distribution is: = 2 /36 + 6/36 + 12/36 + 20/36 + 30/36 + 42/36 + 40/36 + 36/36 + 30/36 + 22/36 +12/36 = 7 The variance 2 is given by: 2 = 25/36 + 32/36 + 27/36 + 16/36 + 5/36 + 0/36 + 5/36 + 16/36 + 27/36 + 32/36 + 25/36 = 212 / 36 = 5.83 = 5.83 = 2.415 Thus when a pair of dice is rolled the expected value of the sum of spots that would appear is 7 with the actual value differing from this expected value to the extent of 2.41 on either side. Varde Varde 10
  • 11. Random Variables and Discrete Distributions
  • 12. Varde Varde 12 Contents. Random Variables Expected Values and Variance Binomial Poisson
  • 13. Varde Varde 13 The discrete r.v The discrete r.v arises in situations when possible outcomes are discrete . Example. Toss a coin 3 times, then S = {HHH,HHT,HTH,HTT, THH, THT, TTH, TTT} Let the variable of interest, X, be the number of heads observed then relevant events would be {X = 0} = {TTT} {X = 1} = {HTT,THT,TTH} {X = 2} = {HHT,HTH, THH} {X = 3} = {HHH}. The relevant question is to find the probability of each these events. Note that X takes integer values even though the sample space consists of Hs and Ts.
  • 14. Varde Varde 14 Discrete Distributions The probability distribution of a discrete r.v., X, assigns a probability p(x) for each possible x such that (i) 0 p(x) 1, and (ii) p(x) = 1 where the summation is over all possible values of x.
  • 15. Varde Varde 15 Discrete distributions in tabulated form Example. Which of the following defines a probability distribution? (i) X 0 1 2 p(x) 0.30 0.50 0.20 (ii) X 0 1 2 p(x) 0.60 0.50 -0.10 (iii) x -1 1 2 p(x) 0.30 0.40 0.20
  • 16. Varde Varde 16 Expected Value and Variance Definition The expected value of a discrete r.v X is denoted by and is defined to be = xp(x). Notation: The expected value of X is also denoted by = E[X]; or sometimes X to emphasize its dependence on X. Definition If X is a r.v with mean , then the variance of X is defned by 2 = (x - )2 p(x) Notation: Sometimes we use 2 = V (X) (or 2 X).
  • 17. Varde Varde 17 Standard Deviation Definition If X is a r.v with mean , then the standard deviation of X, denoted by X, (or simply ) is defined by X = V (X) = (x - )2 p(x)
  • 18. Varde Varde 18 Discrete Distributions-Binomial. The binomial experiment (distribution) arises in following situation: (i) the underlying experiment consists of n independent and identical trials; (ii) each trial results in one of two possible outcomes, a success or a failure; (iii) the probability of a success in a single trial is equal to p and remains the same throughout the experiment; and (iv) the experimenter is interested in the r.v X that counts the number of successes observed in n trials. A r.v X is said to have a binomial distribution with parameters n and p if p(x) = nCx px qn-x (x = 0, 1, . . . , n) where q = 1- p. Mean: = np Variance: 2 = npq, Standard Deviation = v npq
  • 19. Varde Varde 19 Bernoulli. when probability of occurrence of a particular event is constant say p ,the Binomial Distribution gives probabilities of number of occurrences of the event in a series of n trials A r.v X is said to have a Bernoulli distribution with parameter p if n=1 viz only one trial is performed Formula: p(x) = px (1 - p)1-x ; x = 0, 1. Tabulated form: X 0 1 p(x) 1-p p Mean: = p Variance: 2= pq , = v pq
  • 20. Examples of Binomial Situation There are many situations where the outcomes can be grouped into two categories. Binomial distribution is appropriate in describing these situations An employee aspiring for promotion may either be promoted or not promoted, A loan application may either be sanctioned or not sanctioned, Amount advanced may either be recovered or not recovered, A manager may either be retained at HO or may be transferred Indian Captain may either win a toss or lose All these situati


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