Mathematics: GATE2017 EC-2-22

Question

Consider the random process:

$X\left ( t \right )=U+Vt$

where $U$ is zero-mean Gaussian random variable and $V$ is a random variable uniformly distributed between $0$ and $2.$ Assume $U$ and $V$ statistically independent. The mean value of random process at $t=2$ is ___________

Comments

@ankitgupta.1729

Is it in syllabus??:(

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yeah, both gaussian and uniform distribution are in syllabus.

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@ankitgupta.1729

what is $t=2$ means?

yes ans 2, elaborate now

Answer 1

$ X(2) = U + 2V $

$\Rightarrow E[X(2)] = E[ U + 2V] = E[X] + 2 E[V] $

Now, $E[X] = 0$ (given) and $ E[V] = \frac{0+2}{2} = 1$

$\Rightarrow E[X(2)] = 2$

Comments

havenot got this $ E[V] = \frac{0+2}{2} = 1$

how u finding mean??

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For a random variable distributed uniformly between values $a$ and $b$, the mean is $\frac{a+b}{2}$.

Since $V$ is a random variable that is uniformly distributed between $0$ and $2$, It's mean ( or expected value ) = $ \frac{0+2}{2}= 1$

Refer to this link for its derivation: 

https://www.youtube.com/watch?v=hzb7u3lEtHA

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Still not getting , how $E\left [ X\left ( 2 \right ) \right ]=2$