Show that if X has a mgf bounded by the mgf of exponential distribution (i.e. for t 1 we have

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1. Show that if X has a mgf bounded by the mgf of exponential distribution
for t<A), then for Ae> 1 we have
(i.e.
P{X> e} <Aee-(A-1)
(Hint: Use the mgf of exponential distribution to obtain the bound for
P{X> e) then determine its minimum.)
2. Let X have the Poisson distribution with mean λ. Show that
and P {X ≥ 2x} ≤
P{X</}</
3. Let X and Y have the joint density uniform on the triangle with vertices
(0,0), (2,0), and (3, 1).
4
(a) Find E(X/Y) and E(Y/X)
(b) Find Var(X/Y) and var (Y/X)
(c) Find the expectation and variance of X and Y using * and ** (Class
notes)
01
100
Transcribed Image Text:1. Show that if X has a mgf bounded by the mgf of exponential distribution for t<A), then for Ae> 1 we have (i.e. P{X> e} <Aee-(A-1) (Hint: Use the mgf of exponential distribution to obtain the bound for P{X> e) then determine its minimum.) 2. Let X have the Poisson distribution with mean λ. Show that and P {X ≥ 2x} ≤ P{X</}</ 3. Let X and Y have the joint density uniform on the triangle with vertices (0,0), (2,0), and (3, 1). 4 (a) Find E(X/Y) and E(Y/X) (b) Find Var(X/Y) and var (Y/X) (c) Find the expectation and variance of X and Y using * and ** (Class notes) 01 100
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