Let X and Y be independent random variables, each uniformly distributed on the interval [0,2].
Find the mean and variance of XY.
E[XY]=
var[XY]=
Find the probability that XY≥1. Enter a numerical answer.
P(XY≥1)=
E[XY] = 1
var[XY] = 7/9
p(XY >= 1) = ??
p(XY >= 1) = 2/5
p(XY >= 1) = 0.40343
To find the mean and variance of XY, we can use the properties of independent random variables.
Mean (E[XY]):
The mean of the product of independent random variables is simply the product of their individual means. Since X and Y are uniformly distributed on the interval [0, 2], their means are both equal to the average of the interval, which is (0 + 2) / 2 = 1. Therefore, E[XY] = 1 * 1 = 1.
Variance (var[XY]):
To find the variance of XY, we can use the property that the variance of the product of independent random variables is the product of their individual variances.
The variance of a uniformly distributed random variable on the interval [a, b] is given by (b - a)^2 / 12. In this case, since X and Y are uniformly distributed on the interval [0, 2], the individual variances would be (2 - 0)^2 / 12 = 4 / 12 = 1 / 3.
Therefore, var[XY] = var[X] * var[Y] = (1 / 3) * (1 / 3) = 1 / 9.
Probability (P(XY ≥ 1)):
To find the probability that XY ≥ 1, we can utilize the probability distribution of XY.
Since X and Y are both uniformly distributed on the interval [0, 2], the joint probability density function (pdf) of XY is given by f(x, y) = 1 / 4.
To find the probability P(XY ≥ 1), we need to integrate the joint pdf over the region where XY is greater than or equal to 1. Since X and Y are independent, we can integrate them separately.
Integrating with respect to X:
∫[0,2] f(x, y) dx = ∫[0,2] 1 / 4 dx = [x / 4] from 0 to 2 = 2 / 4 - 0 / 4 = 1 / 2.
Integrating with respect to Y:
∫[0,2] f(x, y) dy = ∫[0,2] 1 / 4 dy = [y / 4] from 0 to 2 = 2 / 4 - 0 / 4 = 1 / 2.
Since X and Y are independent, the joint probability distribution is separable, meaning the product of their individual integrals gives the joint probability P(XY ≥ 1):
P(XY ≥ 1) = P(X ≥ 1) * P(Y ≥ 1) = (1 / 2) * (1 / 2) = 1 / 4.
Thus, P(XY ≥ 1) = 1 / 4.