Question

Given a continuous random variable X with a probability distribution function (PDF) defined as follows:

fX(x) = (ax + b) for 0 ≤ x ≤ u

fX(x) = 0 otherwise

where a and b are arbitrary constants, and further, the first moment of the random variable (m1) is k. If u = 4 and m1 (k) = 1.25, determine:

The coefficients a and b in terms of u and k.
The second moment of the random variable (m2).
The standard deviation of the random variable.
The cumulative distribution function (CDF) of the random variable in the range 0 to x, denoted as PX(X).

Answer 1

The student is asked to find the coefficients of a PDF for a continuous random variable, calculate its second moment and standard deviation, and determine its CDF. This involves integral calculus and solving a system of equations to establish the coefficients a and b, then using these to calculate the further required properties of the distribution.

Step-by-step explanation:

The student is asking to determine the coefficients a and b for the probability distribution function (PDF) of a continuous random variable X given the conditions of the function, to find the second moment (m2) of the random variable, and to calculate the standard deviation and the cumulative distribution function (CDF) within a given range. Based on the information provided, we can use the constraints that the first moment (m1) is k, which equals 1.25 in this case, and the upper limit u is 4.

1. We begin by ensuring that the total area under the PDF is equal to 1, which is a fundamental property of PDFs. This is calculated by integrating the given function fX(x) = (ax + b) from 0 to u.
   
2. Because the first moment of X is the expected value, we can integrate x multiplied by the PDF over the range from 0 to u to find a and b such that the result is equal to k (1.25 in this case).
   
3. The second moment (m2) will be obtained by integrating x2 times the PDF over the same range.
   
4. Standard deviation is derived from the square root of the variance, which is m2 minus the square of the first moment (m1).
   
5. The CDF is found by integrating the PDF from 0 to x, where x is any value between 0 and u.
   

To solve for a and b, we set up the equations from the constraints and solve the system of equations. Once a and b are found, we carry on to find m2 and standard deviation using the formulae mentioned above. Calculating the CDF involves integrating the function fX(x) within the limits of the problem and ensuring the result is a function of x.