Distributions

Distributions - a 2 b 7 Density t u(t-4-3-2-1 1 2 0.17 1 3 0.17 2 4 0.17 3 5 0.17 4 6 0.17 5 7 0.17 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

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Unformatted text preview: a 2 b 7 Density t u(t)-4-3-2-1 1 2 0.17 1 3 0.17 2 4 0.17 3 5 0.17 4 6 0.17 5 7 0.17 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 Discrete Uniform Distribution Discrete uniform distribution applies to situations where there are only a finite number of outcomes possible and all of them are equally likely. For instance if you toss a fair coin, there are two outcomes possible: Head or Tail. If we assume that we win $0 if head comes up and $1 if tails come up. Then “ X= the amount of our winnings” is a random variable with two possible values of 0 and 1 each of which is equally likely (and thus with probability ½). If we toss a fair 6-sided die, then the random variable “ X= the number that turns up” which is any of 1, 2, 3, 4, 5, or 6. All of these have equal probability, and thus each has probability 1/6. In general the discrete uniform random variable that can assume any integer from a to b has the probability mass function: Notice that this function is a constant and independent of t because all values have equal probability. The probability distribution function is given by ≤ ≤ +- +- < = . if 1 , integer an is and if 1 1 , if ) , , ( b t t b t a a b a t a t b a t U The expected value of the discrete distribution is the mid point between a and b : E ( X )=( b-a )/2 and Var( X )= a+n/2 . In Excel the distribution function can be calculated by using partial sums. See the formulas in column K after cell K11. 1 1 ) , ; ( +- = a b b a t u 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Cumulative U(t) 0.17 0.33 0.5 0.67 0.83 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1-4-3-2-1 1 2 3 4 5 6 7 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Uniform density function t u(t)-4-3-2-1 0 1 2 3 4 5 6 7 0.2 0.4 0.6 0.8 1 1.2 Uniform Distribution t U(t) Farid Alizadeh: The CDF function can be obtained from the density function by adding all the values upto and including t. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Column J U(t) n p Binomial Distribution For binomial distribution we have two parameters: n which is the number of trials and p which is the probability of “success” at each trial. Then the random variable of interest is: X= ”the number of successes in n trials“. Think of tossing a coin where heads represents “success” and its probability is p. Then the binomial random variable is the number of heads if we toss the coin n times. The formula for probability density function, which is the probability that the binomial random variable has the value t, is given by: t n t p p t n t n p n t b--- = ) 1 ( )! ( ! ! ) , ; ( In this notation, the function b is a function of t; n and p are assumed fixed parameters. The value t can assume any of the values 0, 1, …, n. In the worksheet called “ binomial” we have used the Excel function BINOMDIST to calculate this formula. This function takes 4 arguments. The first three correspond to...
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This note was uploaded on 12/17/2010 for the course MSIS 623:386 taught by Professor Markowitz during the Fall '09 term at Rutgers.

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Distributions - a 2 b 7 Density t u(t-4-3-2-1 1 2 0.17 1 3 0.17 2 4 0.17 3 5 0.17 4 6 0.17 5 7 0.17 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

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