Suppose that W, the amount of moisture in the air on a given day, is a gamma random variable with parameters ( t , β ) . That is, its density is f ( ω ) = β e − β ω ( β ω ) t − 1 Γ ( t ) , ω > 0 . Suppose also that given that W = ω . the number of accidents during that day—call it N—has a Poisson distribution with mean ω . Show that the conditional distribution of W given that N = n is the gamma distribution with parameters ( t + n , β + 1 ) .
Suppose that W, the amount of moisture in the air on a given day, is a gamma random variable with parameters ( t , β ) . That is, its density is f ( ω ) = β e − β ω ( β ω ) t − 1 Γ ( t ) , ω > 0 . Suppose also that given that W = ω . the number of accidents during that day—call it N—has a Poisson distribution with mean ω . Show that the conditional distribution of W given that N = n is the gamma distribution with parameters ( t + n , β + 1 ) .
Suppose that W, the amount of moisture in the air on a given day, is a gamma random variable with parameters
(
t
,
β
)
. That is, its density is
f
(
ω
)
=
β
e
−
β
ω
(
β
ω
)
t
−
1
Γ
(
t
)
,
ω
>
0
. Suppose also that given that
W
=
ω
. the number of accidents during that day—call it N—has a Poisson distribution with mean
ω
. Show that the conditional distribution of
W given that
N
=
n
is the gamma distribution with parameters
(
t
+
n
,
β
+
1
)
.
Consider a real random variable X with zero mean and variance σ2X . Suppose that wecannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and{Wt : t ∈ R} is a WSS process with zero mean and correlation function RW , uncorrelated with X.Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:ˆXT =Z T0h(T − θ)Yθ dθ,i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample theoutput at time T . We wish to design h to minimize the mean-squared error of the estimate.a. Use the orthogonality principle to write down a necessary and sufficient condition for theoptimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)b. Use part a to derive a condition involving the optimal h that has the following form: for allτ ∈ [0, T ],a =Z T0h(θ)(b + c(τ − θ)) dθ,where a and b are constants and c is some function. (You must find a, b, and c in terms ofthe information…
Suppose X, Y, Z are iid observations from a Poisson distribution with parameter λ, which is unknown. Consider the 3 estimators T1 = X + Y − Z, T2 = 2X + Y + Z 4 , T3 = 3X + Y + Z 5 . (a) Which among the above estimators are unbiased? (b) Among the class of unbiased estimators, which has the minimum variance?
1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.
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