A single dot is placed on a very long length of yarn at the textile mill. The yarn is then cut into pieces. The lengths of the different pieces are independent, and the length of each piece is distributed according to the same PDF fX(x). Let R be the length of the piece that includes the dot. Determine the expected value of R in each of the following cases.
In each part below, express your answer in terms of λ using standard notation. Enter 'lambda' for λ.
Suppose that fX(x)={λe−λx,0,x≥0,x<0.
E[R]=
- unanswered
Suppose that fX(x)={λ3x2e−λx2,0,x≥0,x<0.
E[R]=
- unanswered
1)2/lambda
2)4/lambda
CORRECT!!!!!
To determine the expected value of R in each case, we need to find the expected value of the random variable X, which represents the length of each piece.
1. For the first case, where fX(x) = λe^(-λx) for x ≥ 0 and x < 0, we can calculate the expected value as follows:
E[X] = ∫ x * fX(x) dx
= ∫ x * λe^(-λx) dx
To solve this integral, we can use integration by parts. Let u = x and dv = λe^(-λx) dx. Then, du = dx and v = -e^(-λx).
Applying integration by parts, we have:
E[X] = -x * e^(-λx) - ∫ (-e^(-λx)) dx
= -x * e^(-λx) + (1/λ) * e^(-λx)
To find the expected value of R, we need to divide this expression by the probability density function of R. Since R ≥ 0, we have:
E[R] = E[X] / ∫ fX(x) dx
= [-x * e^(-λx) + (1/λ) * e^(-λx)] / ∫ λe^(-λx) dx
= [-x * e^(-λx) + (1/λ) * e^(-λx)] / (1/λ)
= -x * e^(-λx) + (1/λ) * e^(-λx)
Therefore, the expected value of R is -x * e^(-λx) + (1/λ) * e^(-λx).
2. For the second case, where fX(x) = λ^3 * x^2 * e^(-λx^2) for x ≥ 0 and x < 0, we can follow a similar process:
E[X] = ∫ x * fX(x) dx
= ∫ x * λ^3 * x^2 * e^(-λx^2) dx
This integral can be quite challenging to solve analytically. However, one could try numerical integration methods or use computer software to approximate the expected value.
To find the expected value of R, we divide the expression for E[X] by the integral of fX(x):
E[R] = E[X] / ∫ fX(x) dx
Unfortunately, the exact expression for E[R] cannot be determined without solving the integral. Therefore, we cannot provide a specific value for E[R] in this case.