The independent random variables $$\(W\)$$ and $$\(X\)$$ have the following distributions. $$\[ W \sim \operatorname{Po}(4) \quad X \sim \mathrm{B}(3,0.8) \]$$ (a) Write down the value of the variance of $$\(W\)$$ (1) (b) Determine the mode of $$\(X\)$$ Show your working clearly. (2) One observation from each distribution is recorded as $$\(W_{1}\)$$ and $$\(X_{1}\)$$ respectively. (c) Find $$\(\mathrm{P}\left(W_{1}=2\right.\)$$ and $$\(\left.X_{1}=2\right)\)$$ (3) (d) Find $$\(\mathrm{P}\left(X_{1}< W_{1}\right)\)$$ (4)

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