Let X be a continues random variable with PDF (𝑥) = 𝐴𝑒−𝑥3 ; 𝑥 ≥ 0 . Find (i) A (ii) Mean (iii) Variance
Question
Let X be a continues random variable with PDF (𝑥) = 𝐴𝑒−𝑥3 ; 𝑥 ≥ 0 . Find (i) A (ii) Mean (iii) Variance
Solution
To find the values of A, the mean, and the variance, we need to perform the following steps:
(i) Find the value of A: To find the value of A, we need to integrate the probability density function (PDF) over its entire range, which is from 0 to infinity. The integral of the PDF should equal 1, as it represents the total probability of all possible outcomes.
∫[0,∞] A * e^(-x^3) dx = 1
To solve this integral, we need to use techniques such as integration by substitution or numerical methods. Once we solve the integral, we can find the value of A.
(ii) Find the mean: The mean of a continuous random variable can be found by integrating the product of the variable and the PDF over its entire range.
Mean = ∫[0,∞] x * A * e^(-x^3) dx
Again, we need to solve this integral to find the mean.
(iii) Find the variance: The variance of a continuous random variable can be found by integrating the square of the difference between the variable and the mean, multiplied by the PDF, over its entire range.
Variance = ∫[0,∞] (x - Mean)^2 * A * e^(-x^3) dx
Similarly, we need to solve this integral to find the variance.
By following these steps, we can find the values of A, the mean, and the variance for the given continuous random variable X with the given PDF.
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