By Michael Umble, Moukshagundam L. Srikanth

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2. 3. e. what value of a) is this function a bona fide pdf? What is P( X > 5)? Compute the following: (i) Meanof X (ii) Variance of X (iii) Standard Deviation of X (iv) Coefficient of Variation of X (v) Medianof X fx(x) t 3/10 0 10 -X Fig. 5Random variables 49 (v) Median of X From Fig. 12, the modal value is obviously determine the median, one must solve xm 3x2 x" = 10. 94. 3 The hazard function Continuous random variables are often used in risk and reliability analyses. Of particular importance is the time to failure of a component T whose cdf F T ( t ) and pdf f,(t) are typically called the failure probability and density functions at time t.

6: Shaded area represents Event D = A n C 26 4 Basic of Probability Theory for Applications to Reliability and Risk Analysis (ii) E = A uB I I I I r 90" 0 Fig. 7: Shaded area represents Event E = A u B (iii) F=A n B nC v 4 I I : ! 30 I I I I I I I I I I I I 20 I I : Fig. 4 Mutually Exclusive D and E are not mutually exclusive. (Because D nE f 0, in fact D n E = D, ). A and C are not mutually exclusive. ( B e c a u s e A n C g AnC=D). 3 27 Logic of uncertainty: definition of probability As previously explained, for a statement to be an event, it can only have two possible states, either true or false, and at a certain point in time the exact state will become known as a result of the actual perfonning of the associated experiment.

33) Geometric Distribution Considering the previous problem setting of independent trials of the stochastic experiment known as Bernoulli process, we focus now on the probability that the first success occurs at the t - th trial. e. that with all failures in the first t -1 trials (each one occurring with probability 1- p ) and a success at the t-th trial (which occurs with probability p ). The distribution of the corresponding random variable is called Geometric. Its probability mass function is g(t;p ) = (1 - p y lp t = I , 2,.