Suppose X and Y are random variables with joint density

deliriwmfg 2021-11-17 Answered
Suppose X and Y are random variables with joint density function \(\displaystyle{f{{\left({x},{y}\right)}}}={\left\lbrace{0.1}{e}^{{-{\left({0.5}{x}+{0.2}{y}\right)}}}\right.}\) if \(\displaystyle\geq{0},\ {y}\geq{0},{0}\) otherwise. Find the following probabilities. \(\displaystyle{P}{\left({Y}\geq{1}\right)}\)

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Expert Answer

Ceitheart
Answered 2021-11-18 Author has 906 answers
Since \(\displaystyle{f{{\left({x},{y}\right)}}}\) is non-zero only when both x, y are non-negative, we can write
\(\displaystyle{P}{\left({Y}\geq{1}\right)}={\int_{{-\infty}}^{\infty}}{\int_{{{1}}}^{\infty}}{f{{\left({x},{y}\right)}}}{\left.{d}{y}\right.}{\left.{d}{x}\right.}={\int_{{0}}^{\infty}}{\int_{{1}}^{\infty}}{f{{\left({x},{y}\right)}}}{\left.{d}{y}\right.}{\left.{d}{x}\right.}\)
\(\displaystyle={\int_{{0}}^{\infty}}{\int_{{1}}^{\infty}}{0.1}{e}^{{-{\left({0.5}{x}+{0.2}{y}\right)}}}{\left.{d}{y}\right.}{\left.{d}{x}\right.}\)
Use the property \(\displaystyle{n}^{{{a}\cdot{b}}}={n}^{{a}}\cdot{n}^{{b}}\)
\(\displaystyle={0.1}{\int_{{0}}^{\infty}}{\int_{{1}}^{\infty}}{e}^{{-{0.5}{x}}}\cdot{e}^{{-{0.2}{y}}}{\left.{d}{y}\right.}{\left.{d}{x}\right.}\)
\(\displaystyle={0.1}{\left[{\int_{{0}}^{\infty}}{e}^{{-{0.5}{x}}}{\left.{d}{x}\right.}\right]}{\left[{\int_{{1}}^{\infty}}{e}^{{-{0.2}{y}}}{\left.{d}{y}\right.}\right]}\)
\(\displaystyle={0.1}{{\left[-{\frac{{{1}}}{{{0.5}}}}{e}^{{-{0.5}{x}}}\right]}_{{0}}^{\infty}}{{\left[-{\frac{{{1}}}{{{0.2}}}}{e}^{{-{0.2}{y}}}\right]}_{{1}}^{\infty}}\)
\(\displaystyle={0.1}{\left[{\frac{{{1}}}{{{0.5}}}}\right]}{\left[{\frac{{{1}}}{{{0.2}}}}{e}^{{-{0.2}}}\right]}\)
\(\displaystyle={0.1}{\left[{2}\right]}{\left[{5}{e}^{{-{0.2}}}\right]}={e}^{{-{0.2}}}\approx{0.8187}\)
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