Question

Let X1…,Xn be independent random variables, and let W be a random variable such that \({\rm P}\left( {w = c} \right) = 1\) for some constant c. Prove that \({x_1},....,{x_n}\)they are conditionally independent given W = c.

Short answer

\({x_1}....{x_n}\)are conditionally independent given W=c

Step-by-step solution

Random variable      

A random variable is a statistic with an unspecified amount or a function that gives numbers to each of the results of a research.

Compute the probability

n random variables\({x_{1,}}{x_{2,....}}{x_n}\)are called independent if every random variable belongs to the set of real number.

Therefore, we can write:

\({\rm P}\left( {{x_1} \in {A_1}} \right){\rm P}\left( {{x_2} \in {A_2}} \right)....{\rm P}\left( {{x_n} \in {A_n}} \right)\)

Let w be a random variable for some constant c such that\({\rm P}\left( {w = c} \right) = 1\)

Therefore, from the definition of independent random variables, we can say that given w=c \({x_1}....{x_n}\)are conditionally independent.