STA261 - STA261 2A
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STA261 - STA261 2A
1. (2 points) A sample of 6 observations (X1 , ~ ~ ~ , X6 ) is generated from Bernoulli(θ) distribution with θ ∈ [0, 1] unknown but only T = i(6)=1 Xi is observed. De- scribe the statistical model for the observed data.
T - Bin(6, θ), θ ∈ [0, 1] or p← (t) = ╱ 、t(6) θt (1.θ)6.t , t ∈ {0, 1, ~ ~ ~ , 6}, θ ∈ [0, 1]
2. Suppose we have a finite population Π and a measurement X : Π 3 {0, 1} where |Π| = 10 and |{π : X(π) = 0}| = 3.
(a) (2 points) Determine fx (0) and fx (1). Can you identify this population
distribution?
fx (0) = , fx (1) =
This is a binomial distribution Bin(10, 0.7)
(b) (3 points) For a simple random sample of size 4, determine the proba- bility that 4fˆx (0) = 1.
Let X denote the number of elements with measurement 0 in the sam-
ple of size 4, with replacement. X is a hypergeometric random variable. Since fˆx (0) = ,
P (4fˆx (0) = 1) e P (X=1) = = 0.5
(c) (3 points) Under the assumption of i.i.d. sampling, determine the prob- ability that 4fˆx (0) = 1
Let Y denote the number of elements with measurement 0 in the sam- ple of size 4 under i.i.d. sampling. Y is a binomial random variable Bin(4, 0.3).
P (4fˆx (0) = 1) e P (Y=1) = ╱ 、1(4) 0.3 ~ 0.73 = 0.4116
STA261 - Quiz 1B
1. (2 points) A sample of 7 observations (X1 , ~ ~ ~ , X7 ) is generated from Normal(µ, 1) distribution with µ ∈ IR unknown but only T = i(7)=1 Xi is observed. De- scribe the statistical model for the observed data.
T - N (7µ, 7), µ ∈ IR
2. Suppose we have a finite population Π and a measurement X : Π 3 {0, 1} where |Π| = 10 and |{π : X(π) = 0}| = 6.
(a) (2 points) Determine fx (0) and fx (1). Can you identify this population
distribution?
fx (0) = , fx (1) =
This is a binomial distribution Bin(10, 0.4)
(b) (3 points) For a simple random sample of size 3, determine the proba- bility that 3fˆx (0) = 2
Let X denote the number of elements with measurement 0 in the sam-
ple of size 3, with replacement. X is a hypergeometric random variable. Since fˆx (0) = ,
P (3fˆx (0) = 2) e P (X=2) = = 0.5
(c) (3 points) Under the assumption of i.i.d. sampling, determine the prob- ability that 3fˆx (0) = 2
Let Y denote the number of elements with measurement 0 in the sam- ple of size 3 under i.i.d. sampling. Y is a binomial random variable Bin(3, 0.6).
P (3fˆx (0) = 2) e P (Y=2) = ╱ 、2(3) 0.62 ~ 0.4 = 0.432
STA261 - Quiz 1C
1. (2 points) A sample of 5 observations (X1 , ~ ~ ~ , X5 ) is generated from Poisson(λ) distribution with λ > 0 unknown but only T = i(5)=1 Xi is observed. De- scribe the statistical model for the observed data.
λt e.A
T - Poisson(5λ), λ > 0 or pA (t) = t! , t ∈ {0, 1, 2, ~ ~ ~ }, λ > 0
2. Suppose we have a finite population Π and a measurement X : Π 3 {0, 1} where |Π| = 10 and |{π : X(π) = 0}| = 5.
(a) (2 points) Determine fx (0) and fx (1). Can you identify this population
distribution?
fx (0) = , fx (1) =
This is a binomial distribution Bin(10, 0.5)
(b) (3 points) For a simple random sample of size 5, determine the proba- bility that 5fˆx (0) = 0
Let X denote the number of elements with measurement 0 in the sam-
ple of size 5, with replacement. X is a hypergeometric random variable. Since fˆx (0) = ,
P (5fˆx (0) = 0) e P (X=0) = = 0.004
(c) (3 points) Under the assumption of i.i.d. sampling, determine the prob- ability that 5fˆx (0) = 0
Let Y denote the number of elements with measurement 0 in the sam- ple of size 5 under i.i.d. sampling. Y is a binomial random variable Bin(5, 0.5).
P (5fˆx (0) = 0) e P (Y=0) = ╱ 、0(5) 0.50 ~ 0.55 = 0.031
STA261 - Quiz 1D
1. (2 points) A sample of 5 observations (X1 , ~ ~ ~ , X5 ) is generated from Exponential(λ) distribution with λ > 0 unknown but only T = i(5)=1 Xi is observed. De- scribe the statistical model for the observed data.
T - Gamma(5, λ), λ > 0 or pA (t) = e.At , t > 0, λ > 0
2. Suppose we have a finite population Π and a measurement X : Π 3 {0, 1} where |Π| = 10 and |{π : X(π) = 0}| = 5.
(a) (2 points) Determine fx (0) and fx (1). Can you identify this population
distribution?
fx (0) = , fx (1) =
This is a binomial distribution Bin(10, 0.5)
(b) (3 points) For a simple random sample of size 5, determine the proba- bility that 5fˆx (0) = 0
Let X denote the number of elements with measurement 0 in the sam-
ple of size 5, with replacement. X is a hypergeometric random variable. Since fˆx (0) = ,
P (5fˆx (0) = 0) e P (X=0) = = 0.004
(c) (3 points) Under the assumption of i.i.d. sampling, determine the prob- ability that 5fˆx (0) = 0
Let Y denote the number of elements with measurement 0 in the sam- ple of size 5 under i.i.d. sampling. Y is a binomial random variable Bin(5, 0.5).
P (5fˆx (0) = 0) e P (Y=0) = ╱ 、0(5) 0.50 ~ 0.55 = 0.031
2022-02-16