Suppose the reaction temperature X (in °C) in a certain chemical process has a uniform distribution with A = -4 and B = 4. (a) Compute P(X < 0). (b) Compute P(-2 < X < 2). (c) Compute P(-1
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- Suppose the reaction temperature X (in °C) in a certain chemical process has a uniform distribution with A = −6 and B = 6. (a) Compute P(X < 0) (b) Compute P(−3 < X < 3). (c) Compute P(−4≤ X ≤ 5). (Round your answer to two decimal places.)Suppose that the continuous two-dimensional random variable (X, Y ) is uniformly distributed over the square whose vertices are (1, 0), (0, 1), (−1, 0), and (0, −1). Find the Correlation Coefficient ρxySuppose that you enter a fantasy baseball league. Suppose that the 2021 team budget, say , is randomly drawn from a uniform distribution on the interval , where the unit is U.S. million dollars. In addition, suppose that after the value has been observed , the 2022 team budget, say , is randomly drawn from a uniform distribution on the interval . In other words, the 2022 budget is at most as large as the 2021 budget. a) For any given value of x(50<x<350), obtain E[Y|X=x] b) In view of part (a), obtain E[Y|X] c) Atlanta Braves won the 2021 World Series title. Their estimated 2022 payroll is about $130 million. Would your 2022 fantasy baseball budget be on average larger than their 2022 payroll? Explain briefly.
- Suppose that the lifetime in years, X, of a type of electrical switch follows the distribution f(x)=12e−x2f(x)=12e−x2for x>0x>0. What type of continuous distribution is this and what is the average lifetime of one of these switches?Suppose the reaction temperature X (in °C) in a certain chemical process has a uniform distribution with A = −6 and B = 6. (d) For k satisfying −6 < k < k + 4 < 6, compute P(k < X < k + 4). (Round your answer to two decimal places.)Consider two securities, the first having μ1 = 1 and σ1 = 0.1, and the secondhaving μ2 = 0.8 and σ2 = 0.12. Suppose that they are negatively correlated,with ρ = −0.8. Denote the expected return and its standard deviation as functions of π byμ(π ) and σ (π ). The pair (μ(π ), σ (π )) trace out a curve in the plane as πvaries from 0 to 1. Plot this curve in R.
- Suppose that the random variables X1,...,Xn form a random sample of size n from the uniform distribution on the interval [0, 1]. Let Y1 = min{X1,. . .,Xn}, and let Yn = max{X1,...,Xn}. Find E(Y1) and E(Yn).Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0Find the sampling distributions of Y1 and Yn for ran-dom samples of size n from a population having the beta distribution with α = 3 and β = 2.
- LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.At 15:00 it is the end of the school day, and it is assumed that the departure of the students from school can be modelled by a Poisson distribution. On average, 24 students leave the school every minute. (e) There are 200 days in a school year. Given that Y denotes the number of days in the year that at least 700 students leave before 15:30, find (ii) P(Y > 150).If the random variable X follows the uniform distribution U= (0,1) What is the distribution of the random variable Y= -2lnX. Show its limits.