A random variable has the probability density function as Osxs1 ах. 1sxs2 a. f(x) – ax+ 3a , 2< xs3 otherwise
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- b) Let Z₁ =*-~N(0,1), and W₁=~N(0,1), for i=1,2,3,...,10, then: i) State, with parameter(s), the probability distribution of the statistic, T = - 10 ii) Find the mean and variance of the statistic T = iii) Calculate the probability that a statistic T = Z₁ Σας Ζα + W₁ is at most 4.Suppose that a worker value jobs by both the wage rate and the workplace collegiality. The woker’s utility is strictly increasing in the wage rate but strictly decreasing in the chance of being bullied in the workplace. The utility function of the worker is U = wa (1-), where a = 0.5, w is the wage rate and b is the probability that a worker is bullied in a workplace. Suppose for a typical job A, the chance of getting bullied is 0.01, and the wage rate is 100. Which of the following statements is correct? a. If job B offer w = 121 and b = 0.04, the worker would prefer job B to job A b. If job B offer w = 144 and b = 0.09, the worker would prefer job B to job A c. If job B offer w = 80 and b = 0, the worker would prefer job B to job A d. If job B has bullying probablity b = 0.04, the worker is indifferent between job A and job B. Then, the compensating wage differential of job B is at least 25. e. If job B offer w = 80 and b = 0, the worker would be indifferent between job A…If X,, X2, ..., Xn are independent random variables having exponential distributions with the same п parameter 0. (а) Find the probability density of the random variable Y = X1 + X2 + …+ Xn. ... (b) Identify the distribution of Y, and what are its parameters?
- Suppose that the random variable X has the probability density function c(1- - x2) for - 1< x <1 f(x) = { elsewhere What is the variance of X A 1/3 1/5 c) 1/2 1/4В(п 3D 2, р %3D 0.4). Let Y ~ 1. Construct the probability distribution table for Y. P(Y = k) = C"p*(1 – p)" k n- k P(Y = 0) = 1 P(Y = 1) = P(Y = 2) = 2. Use the probability distribution table to find the following: а. Р(Y < - 2) - b. P(Y < 3) c. P(Y < 1) =Exercise 10. For an exponential random variable X (Definition 10), show that E[X] = }. More generally, for a function o : R R, E(¢(X)] = | (u) dF(u) (5.4)
- A random variable X has a probability mass function ƒ(x) = (²) (¹) *, The probability of X ≤ 1may be expressed in lowest term as α P(X ≤ 1) = µ· В . Find a + ß. , x = 1,2,3.Suppose that X is a random variable with density and suppose that ?=e(-4X) Determine ?(?)8. Let the probability density of the random variable ({, n) be f(x, y) = { ke-(2x+7y) x>0 y>o elsewhere Find (1) the constant k; (2) the distribution function of (§, n);
- The daily demand x for water (in millions of gallons) in a town is a random variable with the probability density function (x) = xe 5, [0, ). -x/5, -хе (a) Find the mean and standard deviation of the demand. (Round your standard deviation to three decimal places.) | mean standard deviation (b) Find the probability that the demand is greater than 7 million gallons on a given day. (Round your answer to three decimal places.)Suppose that the random variables X and Y have the following joint probability density function. f(x, y) = ce-8x - 5y, 0 0).Find the probability that the student is female, given that an education degree is not received. P(F|E')