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- The conditional probability of E given that F occurs is P(EF)=___________. So in rolling a die the conditional probability of the event E, getting a six, given that the event F, getting an even number, has occurred is P(EF)=___________.1i. Suppose that a structure can withstand a flood with a peak discharge no greater than 1,950 m3/s and it has been designed to have an economic life span of 100 years. Determine the risk of failure assuming the Extreme Value Distribution when the mean and standard deviation of the annual flood series are 410m3/s and 280 m3/s, respectively. 1ii. What is the probability that at least one flood of ARI 50y (T=50y) will occur during the 30 year design life of a flood control project? 1iii. Consider a small, temporary (3 year design life) flood mitigation dam designed to contain a 20 year flood event. What is the risk that it will be overtopped at least once in the design life.Suppose that the unknown X is uniformly distributed between 0 and 1. What is the expected value of (X^4+2x+1)?
- Suppose a quality control expert examines iterms for defects in a series of independent fixates, each of a fixed duration, and suppose that defects are present. Let p be the probability that the defect or flaw is detected and 1-p the probability that a defect or flaw is not detected. Let the r.v. X = the number of defects detected in n fixations. a. What is the pmf of X? b. What is the expected value of X = E(X)? c. What is the moment generating function (mgf) for the r.v. X (Remember to state the interval of validity for t)? d. Use it to find the Variance of X.If np is greater than or equal to 5 and nq is greater than or equal to 5, estimate P(at least 7) n is equal to 13 and p is equal to 0.6 by using the normal distribution as an approximation to the binomial distribution; if np is less than 5 or nq is less than 5, then state that the normal approximation is not suitableSuppose that you have ten lightbulbs, that the lifetime ofeach is independent of all the other lifetimes, and that eachlifetime has an exponential distribution with parameter l.a. What is the probability that all ten bulbs fail beforetime t?b. What is the probability that exactly k of the ten bulbsfail before time t?c. Suppose that nine of the bulbs have lifetimes that areexponentially distributed with parameter l and thatthe remaining bulb has a lifetime that is exponentiallydistributed with parameter u (it is made byanother manufacturer). What is the probability thatexactly five of the ten bulbs fail before time t?
- A company has 9000 arrivals of Internet traffic over a period of 18,050 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use the formula P(x)= (μ^x • e^−μ) / x! to find the probability of exactly 2 arrivals in one thousandth of a minute, what are the values of μ, x, and e that would be used in that formula?If np≥5 and nq≥5, estimate P(fewer than 3) with n=13 and p=0.4 by using the normal distribution as an approximation to the binomial distribution; if np<5 or nq<5, then state that the normal approximation is not suitable. A. P(fewer than 3)=(blank) (Round to four decimal places as needed.) B. The normal approximation is not suitable.For a random variable X (which is > 0), if logX ~ N(mu, sigma^2), we call the distribution of X "log normal distribution" and write it as X ~ Lognormal(mu, sigma^2) . Show the following: When X ~ Lognoraml(0, 1), the moment generating function of X does not exist.