Question

W represent the wingspan of an airplane and V represents the velocity which are 2 random variables.
W has a normal distribution with mean 10 and standard deviation 4.
Also V = 0.5.W + U, where U is a random variable (we might call error). Assume that U has a standard normal distribution and is independent of W. Derive each element of the variance-covariance matrix for W and V using properties of Variance and Covariance.

Answer 1

Answer :

Given that :

Wingspan of a swallow = W

Velocity = V

Random variable = U

V = 0.5 W + U

i.e W ~ N(10,4)

Therefore E(W) = 10

Standard deviation(W) = = 4

Now given V = 0.5 W + U

Where U ~ N(0,1)

SO,E(U) = 0 and

Standard deviation(U) = V(U) = 1

and

Therefore now consider E(V) = 0.5 E(W) + E(U)

= 0.5 * 10 + 0

= 5

now

In matrix format of W and V is as follows :

Now we need to find COV(WV) = E(W - E(W))(V - E(V))

= E(W - 10)(V - 5)

= E(WV - 5W - 10V + 5V)

Now substitute the values of W and V in above eq

= E(WV) - 5 * 10 - 10 * 5 + 50

= E(WV) - 50

Therefore E(WU) = COV(WV) - 50

E(W(05W + U))

=

= 0.5(116) + 0

= 58

E(WU) = 58

Finally the Variance and Co-variance of matrix is :