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hw7sol - Stat 309 HW7-Chapter 3 1/5 Home Work 7 16.What is...

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Stat 309 HW7-Chapter 3 1/5 Home Work 7 16 .What is the probability density of the time between the arrival of the two packets of Example E in Section 3.4? Ans: Let X = T 1 T 2 Then F X x = P X x = P T 1 T 2 x = P − x T 1 T 2 x = A f T 1, T 2 t 1, t 2 dt 1 dt 2 with A is the area when x T 1 T 2 x , the shaded strip in Figure 3.12. From this example we have area of A is T 2 − T x 2 . Since T 1 ,T 2 are independent uniform random variables on [1,T] we have: f T 1 t 1 = 1 T , t 1 ∈[ 0, T ] and 0 otherwise, f T 2 t 2 = 1 T , t 2 ∈[ 0, T ] and 0 otherwise, f T 1 ,T 2 t 1 ,t 2 = f T 1 t 1 f T 1 t 1 = 1 T 2 , t 1 ∈[ 0, T ] ,t 2 ∈[ 0, T ] and 0 otherwise. Hence: F X x = A 1 T 2 dt 1 dt 2 = 1 T 2 A dt 1 dt 2 = 1 T 2 Area of region A = T 2 − T x 2 T 2 = 1 − 1 x T 2 So probability density of X is f X x = d dx F X x = d dx 1 − 1 x T 2 = 2 T 1 x T 24. Let P have a uniform distribution on [0,1], and, conditional on P=p, let X have a Bernulli distribution with parameter p. Find the conditional distribution of P given X. Ans: By definition of conditional density we have conditional density of P given X is: f P X p x = f X , P x , p f X x So we need to find joint density X, P and marginal density of X. From question we have: P have a uniform distribution on [0,1] so density (pdf) of P is: f P p = 1 , p ∈[ 0,1 ] and 0 otherwise. Conditional on P=p, X has a Bernulli distribution with parameter p so conditional pdf of X given P is: f X P x p = p x 1 p 1 x ,if x = 0 x = 1 Hence by multiplicative law we have joint pdf of X, P: f X , P x , p = f X P x p f P p = p x 1 p 1 x , if x = 0 x = 1, p ∈[ 0,1 ] and 0 otherwise. By definition of marginal density we have: Quoc Tran B248D MSC [email protected]
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Stat 309 HW7-Chapter 3 2/5 f X x = 0 1 f X , P x , p dp = 0 1 p x 1 p 1 x , if x = 0 x = 1 Let x=1 we have f X 1 = 0 1 p 1 p 0 dp = 0 1 p dp =[ p 2 2 ] 0 1 = 1 2 Let x=0 we have f X 0 = 0 1 p 0 1 p dp = 0 1 1 p dp =[ p p 2 2 ] 0 1 = 1 2 so f X x = 1 2 ,if x = 0 x = 1 Conditional density of P given X is: f P X p x = f X , P x , p f X x = p x 1 p 1 x 2 , if x = 0 x = 1, p ∈[ 0,1 ] Noting that Bernulli is special case of binomial distribution, this solution resembles the method in Prof.
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