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

allhvasstH

Answered question

2021-05-16

Let

X_{1} . . . ., X_{n} a n d Y_{1}, . . ., 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

Brittany Patton

Skilled2021-05-17Added 100 answers

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

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