lec5class - CHAPTER 5: Some Discrete Probability...

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CHAPTER 5: Some Discrete Probability Distributions Discrete Uniform Distribution: 5.2 Defnition: If the random variable X assumes the values x 1 , x 2 , . . . , x k with equal probabilities, then the discrete uniform distribution is given by f ( x ; k ) = 1 k , x = x 1 , x 2 , . . . , x k Example: When a die is tossed, each element of the sample space S = { 1 , 2 , 3 , 4 , 5 , 6 } occurs with probability 1/6. Therefore, we have a uniform distribution, with f ( x ; 6) = 1 6 , x = 1 , 2 , 3 , 4 , 5 , 6 . μ = 1 + 2 + ··· + 6 6 = 3 . 5 σ 2 X = 35 12 Binomial Distribution: 5.3 NOTE: Multinomial Distribution is not required. The Bernoulli Process The experiment consists of n repeated trials. Each trial results is an outcome that may be classiFed as a success or a failure. (i.e. heads/tails, correct/erroneous bits, good/defective items, active/silent speakers). The probability of success denoted by p , remains constant from trial to trial. The repeated trials are independent. The number X of successes in n Bernoulli trials is called a binomial random variable . ±or example X could be the number of heads in n tosses of a coin. Example: An early warning detection system for aircraft consists of four identical radar units operating independently of one another. Suppose that each has a probability of 0 . 95 of detecting an intruding aircraft. When an intruding aircraft enters the scene, let X = the number of radar units that do not detect the plane. 1
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The probability distribution of a binomial random variable: Example For the last example P ( X = 3) = P (SSSF, SSFS, SFSS, FSSS) =. The probability distribution of a binomial random variable is called a binomial distribution , and its values will be denoted by b ( x ; n, p ). b ( x ; n, p ) = ± n x ² p x q n - x , x = 0 , 1 , . . . , n where q = 1 - p . Example: Find the probability that seven of 10 persons will recover from a tropical disease if we can assume independence and the probability is 0 . 80 that any one of them will recover from the disease. Solution: Example: Experience has shown that 30% of all persons a±icted by a certain illness recover. A drug company has developed a new medication. Ten people with the illness were selected at random and injected with the medication; nine recover shortly thereafter. Suppose that the medication was absolutely worthless. What is the probability that at least nine of ten injected with the medication will recover?
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lec5class - CHAPTER 5: Some Discrete Probability...

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