Suppose X and Y are random variables with joint density function f(x,y)={0.1e−(0.5x+0.2y) if ≥0, y≥0,0 otherwise. Find the following probabilities. P(Y≥1)

Since f(x,y) is non-zero only when both x, y are non-negative, we can write P(Y≥1)=∫−∞∞∫1∞f(x,y)dydx=∫0∞∫1∞f(x,y)dydx =∫0∞∫1∞0.1e−(0.5x+0.2y)dydx Use the property na⋅b=na⋅nb =0.1∫0∞∫1∞e−0.5x⋅e−0.2ydydx =0.1[∫0∞e−0.5xdx][∫1∞e−0.2ydy] =0.1[−10.5e−0.5x]0∞[−10.2e−0.2y]1∞ =0.1[10.5][10.2e−0.2] =0.1[2][5e−0.2]=e−0.2≈0.8187

Step by step solution to "Suppose X and Y are random variables with..."

deliriwmfg

Answered question

2021-11-17

Suppose X and Y are random variables with joint density function

f (x, y) = {0.1 e^{- (0.5 x + 0.2 y)}

\geq 0, y \geq 0, 0

otherwise. Find the following probabilities.

P (Y \geq 1)

Answer & Explanation

Ceitheart

Beginner2021-11-18Added 14 answers

Since

f (x, y)

is non-zero only when both x, y are non-negative, we can write

P (Y \geq 1) = \int_{- \infty}^{\infty} \int_{1}^{\infty} f (x, y) d y d x = \int_{0}^{\infty} \int_{1}^{\infty} f (x, y) d y d x

= \int_{0}^{\infty} \int_{1}^{\infty} 0.1 e^{- (0.5 x + 0.2 y)} d y d x

Use the property

n^{a \cdot b} = n^{a} \cdot n^{b}

= 0.1 \int_{0}^{\infty} \int_{1}^{\infty} e^{- 0.5 x} \cdot e^{- 0.2 y} d y d x

= 0.1 [\int_{0}^{\infty} e^{- 0.5 x} d x] [\int_{1}^{\infty} e^{- 0.2 y} d y]

= 0.1 {[- \frac{1}{0.5} e^{- 0.5 x}]}_{0}^{\infty} {[- \frac{1}{0.2} e^{- 0.2 y}]}_{1}^{\infty}

= 0.1 [\frac{1}{0.5}] [\frac{1}{0.2} e^{- 0.2}]

= 0.1 [2] [5 e^{- 0.2}] = e^{- 0.2} \approx 0.8187

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