(a) P(X<2) (c) P(4 < X < 8) (b) P(X> 5) (d) P(X<4 or X> 8)
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- 6.) Suppose X is continuously uniformly distributed on [−2, 2]. Let Y = X2. What is the density function of Y? What is the expected value of Y?a. Find the 50-th percentile of X. That is to say the value of x such that P (X ≤ x) = 0.5. b. Now say you have two independent jet engines. What is the probability that only one of themwill last more than 12 months before needing to be rebuilt? c. Find the probability density function f(x) by taking the derivative of F(x) with respect to x.2. Identify the probability density function, then find the mean and variance without integrating. b. f(x) =1/6 e^−x/6, [0,∞) c. f(x) =1 / 3√2π e^−(x−16)^2/18, (−∞,∞)
- Suppose that two continuous random variables X and Y have a joint probability densityfunction f(x, y) = A(x − 3)y for -2≤x≤3 and 4≤y≤6a) What is the value of A?b) What is P(0≤x≤1 and 4≤y≤5)?c) Construct the marginal probability density functions.d) Are the random variables X and Y independent?e) If Y = 5, what is the conditional probability density function of X?f) What are the expectations and variances of the random variables X and Y ?g) What is the covariance of X and Y?h) What is the correlation between X and Y?Suppose that the joint probability density function of X and Y is fX,Y(x,y) = 10.125(x2 – y2) e−3x , for 0<x<∞ and -x<y<x 0, otherwise Give your answers to the below questions in two decimal places where appropriate. (a) The marginal probability density function of X is given by: fX(x) = A xB e-3x , for 0<x<∞ 0, otherwise Find the value of A. (b) Find the value of B. (c) The conditional probability density function of Y, given that X=x for some x>0, takes the following form: fY|X=x(y) = C (x2-y2) xD e-Ex, for -x<y<x 0, otherwise. Find the value of C (d)Find the value of D. (e)Find the value of E.Find a value of k that will make f a probability density function on the indicated interval. ƒ(x) = kx; [1, 5]
- The Weibull distribution has the following density functionf(x) = (α / βα)*xα-1*e-(x/β)^α, y ≥ 0 (Shown in picture) 1. Find the cumulative distribution function and express the median as a function of the parameters α and β.2. Under what condition will this density function equal the exponential density function?3. What would be the expression for the joint distribution of 10 independent random variables with the Weibull density function? Summarize it as much as possible.can you explain how you get P(X=x)xP(Y=y). is this the joint probability mass function?