Joint distribution of (X,Y) depends on unknown parameter 0 and is given by X =1 X =0 0+1 X=-1 2-0 0+1 Y =-1 18 9. 5-0 Y =1 18 6. (a) Find all possible values of parameter 0. -/6
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- 17. In Problem 11 from the previous section, westated that the damage amount is normally distributed. Suppose instead that the damage amount istriangularly distributed with parameters 500, 1500,and 7000. That is, the damage in an accident canbe as low as $500 or as high as $7000, the mostlikely value is $1500, and there is definite skewnessto the right. (It turns out, as you can verify in @RISK,that the mean of this distribution is $3000, thesame as in Problem 11.) Use @RISK to simulatethe amount you pay for damage. Run 5000 iterations. Then answer the following questions. Ineach case, explain how the indicated event wouldoccur.a. What is the probability that you pay a positiveamount but less than $750?b. What is the probability that you pay more than$600?c. What is the probability that you pay exactly $1000(the deductible)?There are two independent random variables X and Y. X has mean 4 and standard deviation , and Y also has mean 4 and standard deviation . A random variable W is the difference between X and Y. That is, W = X - Y. Calculate the mean of W.4.4. An individual picked at random from a population has a propensity to have accidents that is modelled by a random variable Y having the gamma distribution with shape parameter α and rate parameter β. Given Y = y, the number of accidents that the individual suffers in years 1, 2, . . . , n are independent random variables X1, X2, . . . Xn each having the Poisson distribution with parameter y. (a) Write down a function f so that the joint distribution of Y, X1, . . . , Xn can be described via P(a ≤ Y ≤ b, X1 = k1, X2 = k2 . . . Xn = kn) = Z b a f(y, k1, k2, . . . kn)dy and derive from this expression that, for your choice of f, Y has the Gamma distribution, and that conditionally on Y = y, X1, X2, . . . Xn are independent, each having the Poisson distribution with parameter y. (b) Find the conditional distribution of Y given that X1 = k1, X2 = k2, . . . , kn. (c) An insurance company has observed the number of accidents that an individual has suffered on each of n years and wishes to…
- A random variable X is uniform from 4 to 8. A Gaussian random variable Y has mean of 10. Approximate the variance of Y if only 2.5% of its elements is a subset of X.There are two independent random variables X and Y. X has mean 4 and standard deviation , and Y also has mean 4 and standard deviation . A random variable Z is the sum of X and Y. That is, Z = X + Y. Calculate the mean of Z.If Y is an exponential random variable with parameters beta then mean = E(Y) =beta and variance squaared = V(Y) = beta squared. Show proof of this
- A poisson random variables has f(x,3)= 3x e-3÷x! ,x= 0,1.......,∞. find the probabilities for X=0 1 2 3 4 and also find mean and variance from f(x,3).?A random variable X has mean 3 and variance 2. Use Chebyshev’s inequality to obtain an upper bound for P( |X -3| > 2). (answer must be in fraction)Question 29 Suppose that 70% of people prefer beef over lamb. We randomly sampled 1000 people. What is the probability that the sample will have no more than 710 people who prefer beef over lamb? Please use the normal approximation to the probability distribution, and you don't need to use continuity correction here.
- Suppose x has a distribution with μ = 60 and σ = 17. (a) If random samples of size n = 16 are selected, can we say anything about the x distribution of sample means? No, the sample size is too small. Yes, the x distribution is normal with mean μ x = 60 and σ x = 4.25. Yes, the x distribution is normal with mean μ x = 60 and σ x = 17 Yes, the x distribution is normal with mean μ x = 60 and σ x = 1.1. (b) If the original x distribution is normal, can we say anything about the x distribution of random samples of size 16? No, the sample size is too small. Yes, the x distribution is normal with mean μ x = 60 and σ x = 17. Yes, the x distribution is normal with mean μ x = 60 and σ x = 1.1. Yes, the x distribution is normal with mean μ x = 60 and σ x = 4.25. (c) Find P(56 ≤ x ≤ 61). (Round your answer to four decimal places.)Suppose x has a distribution with μ = 60 and σ = 17. (a) If random samples of size n = 16 are selected, can we say anything about the x distribution of sample means? No, the sample size is too small. Yes, the x distribution is normal with mean μ x = 60 and σ x = 4.25. Yes, the x distribution is normal with mean μ x = 60 and σ x = 17 Yes, the x distribution is normal with mean μ x = 60 and σ x = 1.1. (b) If the original x distribution is normal, can we say anything about the x distribution of random samples of size 16? No, the sample size is too small. Yes, the x distribution is normal with mean μ x = 60 and σ x = 17. Yes, the x distribution is normal with mean μ x = 60 and σ x = 1.1. Yes, the x distribution is normal with mean μ x = 60 and σ x = 4.25. Find P(56 ≤ x ≤ 61). (Round your answer to four decimal places.)A sample of 20 observations is taken from the normal distribution where the sample mean is 3. The normal distribution has a mean of 0 and a variance of e^beta. a)Find the estimator for the method of moments for beta and the numerical value of it?