1. Suppose the random variable X represents the failure or not of Edward's speaker woofer while...

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1. Suppose the random variable X represents the failure or not of Edward's speaker woofer while listening to reggae on Friday. Assume there is probability 0.8 that Edward blows up his woofer (woofer fails) while listening to reggae. a. What type of distribution should be used to describe this random experiment? b. What is the probability of the event, Edward does not blow up his woofer? c. Draw the appropriate density function for this distribution and mark appropriately the area under the density function representing probability of the event, Edward does not blow up his woofer. 2. Suppose the random variable X describes a fair 20 sided die. a. What type of distribution should the random variable X follow, so as to describe outcomes of rolls of this die? b. Since distributions generate data, write down 5 possible values that this distribution is likely to generate? c. What is the expectation of X, E[X]? d. What is the variance of X, V[X]? e. Let A = {even outcome}. Use the density function to find the probability of A, P[A]. f. Draw a plot of the density function for this distribution. Mark on the plot the area under the density function representing P[A]. 3. Suppose X ~ Uniform(1, 6) and Y ~ Uniform(1, 20). a. Which random variable has a larger variance, X or Y? b. Explain why your answer to a. makes sense intuitively. Use sentences and/or made up data. 4. Suppose an email message fails to send for 1 out of every 10000 emails you send. a. What's the probability that 1 out of the next 500 emails of yours fails to send? b. What's the probability that at most 1 out of the next 500 emails of yours fails to send? c. What's the probability that at least 2 out of the next 500 emails of yours fail to send? 5. Suppose that over the course of 365 days, 1 million radioactive atoms of Cesium-137 decayed to 977,287 radioactive atoms, giving a decay rate of \(\lambda = (1000000-977287)/365 = 62.23). Use the Poisson distribution to estimate the probability that on a given day, 50 radioactive atoms decayed. 6. Let X ~ Binomial (3, p). For what value of p is f(2|3, p) maximized? 1. Suppose the random variable X represents the failure or not of Edward's speaker woofer while listening to reggae on Friday. Assume there is probability 0.8 that Edward blows up his woofer (woofer fails) while listening to reggae. a. What type of distribution should be used to describe this random experiment? b. What is the probability of the event, Edward does not blow up his woofer? c. Draw the appropriate density function for this distribution and mark appropriately the area under the density function representing probability of the event, Edward does not blow up his woofer. 2. Suppose the random variable X describes a fair 20 sided die. a. What type of distribution should the random variable X follow, so as to describe outcomes of rolls of this die? b. Since distributions generate data, write down 5 possible values that this distribution is likely to generate? c. What is the expectation of X, E[X]? d. What is the variance of X, V[X]? e. Let A = {even outcome}. Use the density function to find the probability of A, P[A]. f. Draw a plot of the density function for this distribution. Mark on the plot the area under the density function representing P[A]. 3. Suppose X ~ Uniform(1, 6) and Y ~ Uniform(1, 20). a. Which random variable has a larger variance, X or Y? b. Explain why your answer to a. makes sense intuitively. Use sentences and/or made up data. 4. Suppose an email message fails to send for 1 out of every 10000 emails you send. a. What's the probability that 1 out of the next 500 emails of yours fails to send? b. What's the probability that at most 1 out of the next 500 emails of yours fails to send? c. What's the probability that at least 2 out of the next 500 emails of yours fail to send? 5. Suppose that over the course of 365 days, 1 million radioactive atoms of Cesium-137 decayed to 977,287 radioactive atoms, giving a decay rate of \(\lambda = (1000000-977287)/365 = 62.23). Use the Poisson distribution to estimate the probability that on a given day, 50 radioactive atoms decayed. 6. Let X ~ Binomial (3, p). For what value of p is f(2|3, p) maximized?