Question

If \(X\), \(Y\) and \(Z\) are independent and identically distributed (i.i.d.) random variables, then the mean and variance of \(X\), \(Y\) and \(Z\) are

• A. Mean and variance are same for \(X\), \(Y\) and \(Z\)
• B. Mean and variance are different for \(X\), \(Y\) and \(Z\)
• C. Mean is same but variance is different for \(X\), \(Y\) and \(Z\)
• D. Mean is different but variance is same for \(X\), \(Y\) and \(Z\)

Hint:

For i.i.d. random variables: \[ \text{Same Distribution} \] \[ \Rightarrow \] \[ \text{Same Mean} \] and \[ \text{Same Variance} \] The word “identically” is the key clue.

Answer 1

The Correct Option is A

Concept: The abbreviation i.i.d. stands for: \[ \boxed{\text{Independent and Identically Distributed}} \] This is one of the most important concepts in probability and statistics.

Step 1: Understand the term Independent.
Independent random variables do not influence each other's outcomes. Mathematically, \[ P(X,Y)=P(X)P(Y) \] for independent variables.

Step 2: Understand the term Identically Distributed.
Identically distributed means all variables follow the same probability distribution. Therefore: \[ X,\ Y,\ Z \] have the same distribution.

Step 3: Relate distribution to mean and variance.
Since all three variables follow the same distribution: \[ E(X)=E(Y)=E(Z) \] Similarly, \[ Var(X)=Var(Y)=Var(Z) \]

Step 4: Determine the correct option.
Both the mean and variance are identical for all three random variables. \[ \boxed{ E(X)=E(Y)=E(Z) } \] and \[ \boxed{ Var(X)=Var(Y)=Var(Z) } \] Hence option (A) is correct.