Chapter 6: Q.6.10 (page 271)

$T h e j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n o f X a n d Y i s g i v e n b y f (x, y) = e - (x + y) 0 \dots x < q, 0 \dots y < q F i n d (a) P X < Y a n d (b) P X < a$

Short Answer

Expert verified

(a) $P (X < Y) = \frac{1}{2}$

$(b) P (X < a) = 1 - e^{- a}$

Step by step solution

Introduction

The joint probability density function of X and Y $f (x, y) = e - (x + y) 0 \dots x < q, 0 \dots y < q$

Explanation of (a).

G i v e n i n f o r m a t i o n : T h e j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n i s f (x, y) = e^{- (x + y)} \leq x \leq \infty, 0 \leq y \leq \infty formula used: P (X < Y) = \int_{0}^{\infty} \int_{0}^{y} f (x, y) d x d y \int_{0}^{\infty} \int_{0}^{y} e^{- x} \times e^{- y} d x d y \int_{0}^{\infty} {[- e^{- x}]}_{0}^{y} \times e^{- y} d y \int_{0}^{\infty} (- e^{- y} + 1) \times e^{- y} d y \int_{0}^{\infty} (e^{- y} - e^{- 2 y}) d y S o l v i n g i t f u r t h e r {[- e^{- y} - \frac{e^{- 2 y}}{- 2}]|}_{0}^{\infty} = 1 - \frac{1}{2} = \frac{1}{2}

Explanation of (b).

Formula used:

P (X < a) = \int_{0}^{\infty} \int_{0}^{a} f (x, y) d x d y C a l c u l a t i o n : F i n d t h e r e q u i r e d p r o b a b i l i t y u \sin g t h e a b o v e f o r m u l a \int_{0}^{\infty} \int_{0}^{a} e^{- x} \times e^{- y} d x d y \int_{0}^{\infty} {[- e^{- x}]}_{0}^{a} \times e^{- y} d y \int_{0}^{\infty} (- e^{- a} + 1) \times e^{- y} d y (- e^{- a} + 1) \times {\frac{e^{- y}}{- 1}|}_{0}^{\infty} = 1 - e^{- a}

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$T h e j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n o f X a n d Y i s g i v e n b y f (x, y) = e - (x + y) 0 \dots x < q, 0 \dots y < q F i n d (a) P X < Y a n d (b) P X < a$

Short Answer

Step by step solution

Introduction

Explanation of (a).

Explanation of (b).

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ThejointprobabilitydensityfunctionofXandYisgivenbyf(x,y)=e−(x+y)0…x<q,0…y<qFind(a)PX<Yand(b)PX<a

Short Answer

Step by step solution

Introduction

Explanation of (a).

Explanation of (b).

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$T h e j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n o f X a n d Y i s g i v e n b y f (x, y) = e - (x + y) 0 \dots x < q, 0 \dots y < q F i n d (a) P X < Y a n d (b) P X < a$