Expert answers to "Let X1….,XnandY1,…,Ym be two sets of random variables...."

EunoR

Answered question

2021-11-06

Let

X_{1} \dots ., X_{n} and Y_{1}, \dots, Y_{m}

be two sets of random variables. Let

a_{i}, b_{j}

be arbitrary constant.
Show that

C o v (\sum_{i = 1}^{n} a_{i} X_{i}, \sum_{j = 1}^{m} b_{j} Y_{j}) = \sum_{i = 1}^{n} \sum_{j = 1}^{m} a_{i} b_{j} C o v (X_{i}, Y_{j})

Answer & Explanation

Mitchel Aguirre

Skilled2021-11-07Added 94 answers

Step 1:
It is given that $X_{1}, \dots, X_{n} and Y_{1}, \dots, Y_{m}$ be the two sets of random variables.
Let $a_{i}, b_{j}$ be the arbitrary constant.
It is ed to show,
$C o v (\sum_{i = 1}^{n} a_{i} X_{i}, \sum_{j = 1}^{m} b_{j} Y_{j}) = \sum_{i = 1}^{n} \sum_{j = 1}^{m} a_{i} b_{j} C o v (X_{i}, Y_{j})$
Step 2
Proof :
$C o v (\sum_{i = 1}^{n} a_{i} X_{i}, \sum_{j = 1}^{m} b_{j} Y_{j})$
$= E (\sum_{i = 1}^{n} a_{i} X_{i}, \sum_{j = 1}^{m} b_{j} Y_{j}) - E (\sum_{i = 1}^{n} a_{i} X_{i}) E (\sum_{j = 1}^{m} b_{j} Y_{j})$
$= E \sum_{i = 1}^{n} \sum_{j = 1}^{m} a_{i} b_{j} [E (X i Y j) = E (X i) E (Y j)]$
$= E \sum_{i = 1}^{n} \sum_{j = 1}^{m} a_{i} b_{j} C \otimes (X i, Y j) \dots . b y c o v a r i a n c e \propto e r t y C o v (X, Y) = E (X Y) - E (X) E (Y)$ .
Hence , It is proved.

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