(1)本問を選択

(1)

確率・統計

(1)

本問を選択

(Select this problem){

する

(Yes)

，しない

(No) } No.

ある離散型の確率変数

X

の確率母関数

g(t) = E(t

^X

)

が以下で与えられる。

(The probability generating function of a certain discrete random variable X is given by the following expression g(t):)

g(t) = exp(λ(t − 1))

1.

この分布の平均と分散を求めよ。

(Obtain the mean and variance of this distribution.)

2. g(t)

を

t

のベキに展開し、この分布を求めよ。

(By expanding g(t) in power of t, identify the distribution.)

(2)

確率・統計

(2)

本問を選択

(Select this problem){

する

(Yes)

，しない

(No) } No.

正規母集団

N (µ, 25)

から大きさ

n

の無作為標本

X

₁

, X

₂

, ..., X

_nを得たとする。また、確率変数

Z

が標準正規分布に従うとき、

P (Z > 1.282) = 0.10, P (Z > 1.645) = 0.05

とする。次の問いに答えよ。

( Let X = {X

₁

, X

₂

, · · · , X

_n

} be a random sample of size n from a normal distribution N(µ, 25). Note that P (Z > 1.282) = 0.10, P (Z > 1.645) = 0.05 if Z has a standard normal distribution. Answer the following questions:)

1. P (|Z| > 1.282)

を求めよ。

(Compute the probability P (|Z| > 1.282). )

2.

標本平均

X ¯

_n

=

¹_n^Pⁿ_i=1

X

_i の確率分布を求めよ。ヒント：正規分布は和や定数倍について再生性をもつ。

( Obtain the probability distribution of the sample mean ¯ X

_n

=

¹_n^Pⁿ_i=1

X

_i

. Hint: Normal distribution is reproductive under summation and constant multiplication. )

3.

母平均

µ

の

90%

信頼区間を求めよ。

(Find a 90 percent confidence interval for µ.)

4.

母平均

µ

の

90%

信頼区間幅が

5.0

未満となる最小の

n

を求めよ。

(1)本問を選択

(1)

(Select this problem){

(Yes)

(No) } No.

X

g(t) = E(t

)

(The probability generating function of a certain discrete random variable X is given by the following expression g(t):)

g(t) = exp(λ(t − 1))

1.

(Obtain the mean and variance of this distribution.)

2. g(t)

t

(By expanding g(t) in power of t, identify the distribution.)

(2)

(Select this problem){

(Yes)

(No) } No.

N (µ, 25)

n

X

, X

, ..., X

Z

P (Z > 1.282) = 0.10, P (Z > 1.645) = 0.05

( Let X = {X

, X

, · · · , X

} be a random sample of size n from a normal distribution N(µ, 25). Note that P (Z > 1.282) = 0.10, P (Z > 1.645) = 0.05 if Z has a standard normal distribution. Answer the following questions:)

1. P (|Z| > 1.282)

(Compute the probability P (|Z| > 1.282). )

2.

X ¯

=

X

( Obtain the probability distribution of the sample mean ¯ X

=

X

. Hint: Normal distribution is reproductive under summation and constant multiplication. )

3.

µ

90%

(Find a 90 percent confidence interval for µ.)

4.

µ

90%

5.0

n

(Find minimum n such that width of

90 percent confidence interval for µ is less than 5.0.)