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- The fire department of a city wants to test the null hypothesis that σ =10 minutes for the time it takes a fire truck to reach a fire site against thealternative hypothesis σ 6= 10 minutes. What can it conclude at the 0.05 levelof significance if a random sample of size n = 48 yields s = 9.5minutes?The fire department of a city wants to test the null hypothesis that σ = 10 minutes for the time it takes a fire truck to reach a fire site against the alternative hypothesis σ 6= 10 minutes. What can it conclude at the 0.05 level of significance if a random sample of size n = 30 yields s = 9.5 minutes? Assume normality.Suppose that the random variable X follows a beta distribution with alpha=1 and beta=3, Beta(1,3) Find p(x>1/3) In R simulate n = 1000 from Beta(1,3) p(x>1/3) and verify the probability is close to the theoretical.
- The proportion of snacks in the pantry that will be consumed between now and the next grocery store trip is modeled as a beta random variable on [0,1] with parameters α=1 and β=2. a) What is the probability that at least 50% of the snacks will be consumed? b) What is the probability that at most 10% of the snacks will be consumed? c) What is the expected value of the proportion that will be consumed?Let X be a continuous random variable, normally distributed with a mean of 46 and a standarddeviation of 4.2. What value of x would have a z-score of 1.56?The number of times that a person has a cold each year is a Poisson random variable with parameter λ = 5. Suppose that a new drug has been developed that reduces this to λ = 3, but that the drug only works for 75% of the population. For the remaining 25% of the population there is no effect. A person takes the drug for an entire year. During the year the person had 2 colds. What is the probability that the drug had an effect on this person?
- For an initial investment of 100, an investment yields returns of Xi at the end of period i for i = 1,2, where X1 and X2 are independent normal random variables with mean 60 and variance 25. What is the probability the rate of return of this investment is greater than 10 percent?A random variable X is uniform from 4 to 8. A Gaussian random variable Y has mean of 10. Approximate the variance ofYY if only 2.5% of its elements is a subset of X.Suppose X is a random variable with standard beta distribution and parameters (alpha) = 8 and (beta) = 2 What is (PX>=0.6)?
- Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with parameter λ=3.1i. Find the probability that 3 or more accidents occur today ii. Find the probability that no accident occur todayFor an initial investment of 100, an investment yields returns of Xi at the end of period i for i=1, 2 where X1 and X2 are independent normal random variables with mean 60 and variance 25. What is the probability the rate of return of this investment is greater than 10%?Let us consider a discrete random variable having the pmf given by, PX(k) ={ (1/3) , k = 1 (2/3) , k = 2. 0 , k = 3 Calculate the moment generating function for X. Also get the values of its mean and variance.