[Q3] The continuous joint probability distribution is expressed P(X <3, Y < 4). (Work at precision of 4 digits after the decimal point.) (cu²e for 2 < x < 4, 3 < y < 5
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- Assume that the probability that an airplane engine will fail during a torture test is 12and that the aircraft in question has 4 engines. Construct a sample space for the torture test. Use S for survive and F for fail.If the joint probability distribution of X1 and X2 isgiven by f(x1, x2) = x1x236for x1 = 1, 2, 3 and x2 = 1, 2, 3, find(a) the probability distribution of X1X2;(b) the probability distribution of X1/X2.Assume you have created a 2-stock portfolio by investing $30,000 in stock X with a beta of 0.8, and $70,000 in stock Y with a beta of 1.2. Market risk premium is 8% and risk-free rate is 6%. The followings are the probability distributions of Stocks X and Y’s future returns: State of Economy Probability rx rY Recession 0.1 -10% -35% Below average 0.2 2% 0% Average 0.4 12% 20% Above average 0.2 20% 25% Boom 0.1 38% 45% Calculate the portfolio’s expected rate of return and the standard deviation of its future returns Calculate the required rate of return of your portfolio. Which stock in…
- Determine f(2) so that the given function can serve as a probability distribution of the discrete random variable X: f(x) = k(x2 + 4), for x = 0, 1, 2, 3. Use 4 decimal places.The probability distribution of random variable X is defined as follows: X 0 1 2 3 4 Probability 0 0.3 0.1 0.3 0.3 We transform X using function g(X) =4X2 + 2. What is expected value and standard deviation of g(X), that is, what is E[g(X)] and σ[g(x)]?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.
- The probability function of the random variable X is defined as f(x)=cx²(1-x)² for 0<x<1, otherwise f(x)= 0. Calculate the constant c , the expected value and the variance.The distribution of the random variable X is determined by the function: f(x) = ((x-1)2)/c ; for 1<x<3 = 0 ; inak Sketch the graph of f(x) Determine the FY(y) and fY(y) for Y = X^2 - 1(a) Let F denote the cumulative distribution function (cdf) of a uniformly distributed random variable X. If F(2) = 0.3, what is the probability that X is greater than 2 ? (b) Let F denote the cdf of a uniformly distributed random variable X. If F(2) = 0.3, and F(3) = 0.6, what is F(6) ? (c) Suppose X and Y are Poisson Random Variables. X has a mean of 1 and Y has a mean of 2. X and Y are correlated with CORR (X,Y)=0.5. whats the variance of X+Y
- For the continuous probability function f(x ) = kx^2e^-x when 0≤x≤1. Find (a)k (b)mean (c)varianceUse the moment generating function technique to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Exponential(θ), for i =1, . . . , n. Find the distribution of Y = X1 + · · · + Xn.f(x) for a continuous probability function is 1/18 , and the function is restricted to 2 ≤ x ≤ 20. What is P(x < 2)?