By Chien W., Hung Q.
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Additional info for 3-D numerical study on the bending of symmetric composite laminates
Unlike in the discrete probability case, there is no simple analytical formula for obtaining the probability for the normal distribution. Instead, in evaluating such a probability, we often consider the transformation Z = (X − µ)/σ. It can be shown that when X ∼ N(µ, σ 2 ), then Z = X−µ ∼ N(0, 1), σ which is a normal distribution with mean 0 and variance 1. The particular N(0, 1) distribution is called the standard normal distribution. 1) when X ∼ N(70, 62 ). 1 under the probability density curve (note that the total area under any density curve is 1).
A and B) is just the event that the value shown is a 2. Thus, P(AB) = 1/6. 6) is 3/6 + 3/6 − 1/6 = 5/6, which is equal to P(A or B) = 5/6 that we obtained earlier. 2 DEPENDENT EVENTS AND CONDITIONAL PROBABILITY 9 or generally with n events E1 , . . 7) i=1 where the notation ki=1 ai stands for the summation of the terms ai over the index i from 1 to k, n n P i=1 Ei means P(E1 ∩ E2 ∩ · · · ∩ En ) i=1 Ei means P(E1 ∪ E2 ∪ · · · ∪ En ), and P or just P(E1 E2 · · · En ). 7) is also called the principle of inclusion and exclusion.
3. 7528. 4115 are obtained from the standard normal probability table given in Appendix A. The first probability P(−1 < Z < 0) is obtained based on the fact that the standard normal distribution is symmetric about its mean 0. 9 Likelihood ratio Consider a blood donation center in which every donor is required to have a screening blood test for a particular kind of disease. Suppose a person has had the test and the result is positive. In this situation, two propositions or explanations may be considered: H0 : the person is a carrier of the disease; H1 : the person is not a carrier of the disease.
3-D numerical study on the bending of symmetric composite laminates by Chien W., Hung Q.