Consider a model involving multiple observations of the form X_ i=c_ i\Theta +W_ i, i=1,2,\ldots ,n, where \Theta ,W_1,\ldots ,W_ n are independent (not necessarily normal) random variables and the c_ i's are known nonzero constants. Assume that \Theta has positive variance.
a) Are the random variables X_ i, i=1,2,\ldots ,n, independent?
b) Are the random variables X_ i, i=1,2,\ldots ,n, conditionally independent given \Theta?
a) The random variables X_i, i=1,2,...,n, are not independent. This is because they all have a common component, the variable Θ. Any change or variation in Θ would affect all the X_i variables, which means there is a dependence among them.
b) The random variables X_i, i=1,2,...,n, are conditionally independent given Θ. This means that once the value of Θ is known, the X_i variables become independent of each other. However, without knowing the value of Θ, the X_i variables are not independent.
a) No, the random variables X_i, i=1,2,...,n, are not independent. This is because they are all dependent on the random variable Theta. In other words, the value of Theta influences the values of X_i for all i. Therefore, the random variables X_i are dependent on each other.
b) Yes, the random variables X_i, i=1,2,...,n, are conditionally independent given Theta. This means that once we know the value of Theta, the values of X_i for different i's are independent of each other. However, without knowing the value of Theta, the variables X_i are dependent on each other.