1. No category

Assignment-6
20.04.10
1. Let X1 , X2 be random sample from N (0, 1)
(a) What is the distribution of
(X2√
−X1 )
?
2
(b) What is the distribution of
(X1 +X2 )2
?
(X2 −X1 )2
(c) What is the distribution of √(X1 +X2 ) 2 ?
(X1 −X2 )
(d) What is the distribution of
1
Z
if Z =
X12
?
X22
2. Let Z1 , Z2 be a random sample of size 2 from N (0, 1) and X1 , X2 be random
sample of size 2 from N (1, 1). Suppose that Zi s are independent of the Xj s.
(a) What is the distribution of X + Z?
(b) What is the distribution of √
(Z1 +Z2 )
[(X2 −X1 )2 +(Z2 −Z1 )2 ]/2
?
(c) What is the distribution of [(X1 − X2 )2 + (Z1 − Z2 )2 + (Z1 + Z2 )2 ]/2?
(d) What is the distribution of
(X2 +X1 −2)2
?
(X2 −X1 )2
3. Suppose X1 , X2 , X3 , X4 , X5 are iid N (0, 1). Determine the constant K such that
the r.v.
K(X1 + X2 )
q
(X32 + X42 + X52 )
will have t− distribution.
4. Let X1 , X2 iid from N (0, σ 2 ). Determine the value of
(X1 + X2 )2
P
<4 .
(X1 − X2 )2
#
"
5. Suppose that a point (X,Y,Z) is chosen at random in three- dimensional space,
where X,Y, and Z are independent r.vs. and each has standard normal distribution. What is the probability that the distance from the origin to the point
will be less than 1 unit.
6. Let X1 , . . . , Xn ∼ N(0,1). Find the distribution of
n(Xn −µ)2
.
σ2
7. Given 10 iid r.vs. Xi , where Xi ∼ N (0, 25) for i = 1, 2, . . . , 10. Find the number
k such that
v

u
10
u 1 X
P t
X 2 ≥ k  = 0.5
10 i=1 i
8. The tensile strength (X) for a type of wire is normally distributed with an
unknown mean µ and unknown variance σ 2 . Five pieces are randomly selected
from a large roll, and strength of each segment of the wire is measured. Find the
probability that the sample mean X will be within √2Sn of the true population
mean µ. ( S 2 is sample variance).
1
9. Let S 2 be the sample variance of a random sample of size 26 from N(µ, 16). The
value of P (9.35 < S 2 < 24.09) is ...............
10. Let X1 , . . . , Xn be a random sample of size n from normal distribution with
1 Pn
1 Pn
2
2
2
mean µ and variance σ 2 . Let S 2 = n−1
i=1 (Xi −X) and S0 = n
i=1 (Xi −X) .
Find E(S 2 ), E(S02 ), V (S 2 ) and V (S02 ).
11. A random sample of size 11 is taken from a N (µ, σ 2 ) population where both µ
2
and σ 2 are unknown and the S 2 is computed. Compute the P (0.487 < Sσ2 <
1.599).
12. Let X1 , X2 , X3 , X4 , X5 , X6 be a random sample from a N (0, σ 2 ) population.
2 +X3
follows a t3 −
Find the value of constant K so that the statistic K √X1 +X
2
2
2
X4 +X5 +X6
distribution.
13. A certain type of thread is manufactured with mean tensile strength of 78.3 kg
and a standard deviation of 5.6 kg. How is the variance of the sample mean
changed when the sample size is increased from 64 to 196?
14. If S12 and S22 represent the variances of independent random samples of size
n1 = 8 and n2 = 12, taken from two normal populations with equal variances,
S2
find P ( S12 < 4.89).
2
15. Suppose that a r.v. X can take only five values x=1,2,3,4,5 with following
probabilities:
f (1| θ) = θ3 , f (2| θ) = θ2 (1 − θ), f (3| θ) = 2θ(1 − θ), f (4| θ) = θ(1 −
θ)2 , f (5| θ) = (1 − θ)3 , 0 ≤ θ ≤ 1.
Consider an unbiased estimator Uc (X) such that
Uc (1) = 1, Uc (2) = 2 − 2c, Uc (3) = c, Uc (4) = 1 − 2c, Uc (5) = 0. For what
value(s) of c, Uc (X) is unbiased estimator of θ.
16. Show that the sample variance S12 =
of population variance.
1
n
¯ 2 is not an unbiased estimator
(Xi − X)
P
17. Let X1 , . . . , Xn be a random sample of size n from Poisson distribution with
2
mean λ. Is X an unbiased estimator of λ2 ? Is it an asymptotically unbiased
estimator of λ2 ?
18. The random variable (r.v.) X can take on values 0, 1, 2, and 3 with probabilities
p3 , (1 − p)p2 , (1 − p)2 and 2p(1 − p), where 0 < p < 1, respectively. Find the
maximum likelihood estimator (MLE) of p if a random sample of size n = 150
resulted in a 0 twenty-four times, a 1 fifty-four times, a 2 thirty-two times, and
a 3 forty times.
19. The time a client waits to be served by the mortgage specialist at a bank has
density function
1
f (x, θ) = 3 x2 e−x/θ x > 0, θ > 0
2θ
2
(a) Derive the maximum likelihood estimator of θ for a random sample of size
n. Is the estimator unbiased. Find the variance of this estimator.
(b) Derive the method of moment estimator of θ.
(c) If the waiting times of 15 clients are 6, 12, 15, 14, 12, 10, 8, 9, 10, 9, 8, 7,
10, 7 and 3 minutes, compute the maximum likelihood estimate of θ.
20. Let X1 , . . . , Xn be a random sample from
fX (x, θ) = θ(1 + x)−(1+θ) , θ > 1, x > 0.
Find the moment estimator of θ.
21. Given a random sample of size n from a Gamma(r, λ) population, find the
method of moments estimator of r and λ.
22. Let X1 , . . . , Xn be a random sample from exponential distribution with parameter θ (unknown). Determine the MLE of median of the distribution.
23. Let X1 , . . . , Xn be a random sample of size n from Poisson distribution with
mean λ. Obtain the MLE of P (X = 0).
24. Let X1 , . . . , Xn be a random sample of size n from the distribution with pdf
f (x, µ, σ) =
1
√
xσ 2π
1
2
e− 2σ2 (lnx−µ) ,
x ≥ 0, −∞, µ < ∞, σ > 0.
Find the MLE for mean and variance of X.
Note that
σ2
2
2
E(X) = eµ+ 2 , V (X) = e2µ+σ (eσ − 1).
25. Let X1 , . . . , Xn be a random sample of size n from the distribution with pdf
from
1
c − θ < x < c + θ; 0 otherwise.
f (x; c, θ) =
2θ
Use the method of moments to estimate c and θ.
26. Let X ∼ G(r, λ). Obtain a sufficient statistic for the parameter λ if r is known.
27. Prove that M SE(T ) = V ar(T ) + (Bias)2
28. (a) State the Factorization Theorem.
(b) Let X1 , . . . , Xn be a random sample of size n from the distribution with
pdf
f (x; λ, µ) =
1
xλ−1 (1 − x)µ−1 0 < x < 1, λ > 0, µ > 0; 0 otherwise.
B(λ, µ)
i. If λ is known, find sufficient statistic for µ.
ii. If µ is known, find sufficient statistic for λ.
3
29. Suppose the time “X” (in hours) between orders for a given part at a large
warehouse is gamma random variable with parameters r and λ. Find by the
method of moment the estimate of r and λ based on the sample data
15.5, 4.5, 6.8, 46.0, 34.5, 4.7, 20.9, 8.2, 14.9, 17.7.
30. Let X1 , . . . , Xn be a random sample of size n taken from lognormal pdf
f (x; µ, σ) =
1
√
xσ 2π
exp{−
1
(lnx − µ)2 }, x > 0.
2σ 2
Find the method of moment estimator for µ and σ 2 .
31. Let X1 , . . . , Xn be a random sample of size n from the following distributions
with pmf(pdf) given. In case obtain MLE of the parameter(s) as mentioned.
(a)
f (x) =
α α−1 (−x/β)α
x e
x>0
βα
Obtain MLE of α and β.
(b)
f (x) =
2x −x2 /α2
e
x>0
α2
Obtain MLE of α.
(c) Let X ∼ N (θ, θ). Obtain MLE of θ.
32. Suppose X1 , X2 , .....Xn is a random sample from the probability density function
f (x, λ) , where f (x, λ) = λx2 exp{− λx }, x > 0, λ > 0.
Find the maximum likelihood estimator of λ.
If the expected value of this density function is 2λ. Do you think your estimator
is an unbiased one?
33. Let X ∼ U(0, θ). It is decided to test H0 : θ = 2 vs H1 : θ 6= 2 on the basis of a
random sample of size 1 using the following rule: Reject H0 if either x1 ≤ 0.1
or x1 ≥ 1.9. Find α. Also find β if the true value of θ is 2.5.
34. Let X ∼ P (θ) and sample size n = 12. For testing H0 : θ = 1/2 V s H1 : θ =
P
1/4, suppose the test is: reject H0 if 12
i=1 ≤ 2. Find α and β.
35. Let X ∼ N (µ, 9) and sample size n = 12. For testing H0 : µ = 15 V s H1 :
µ = 20, suppose the test is: reject H0 if x ≤ c. For α = 0.05, determine the
value of c. Also find β.
36. Let Let X ∼ B(1, p), where p is the probability of tossing a head. It is desired
to test H0 : p = 0.5 V s H1 : p = 0.7. A coin is tossed 15 times. Let Y equals
the number of times a head is observed in 15 tosses of this coin. Assume the
rejection region to be Y ≥ 10. Find α and β, respectively.
4
37. Independent random samples of sizes n1 = 100 and n2 = 100 were selected from
N(µ1 , σ12 ) and N(µ2 , σ22 ), respectively. The samples yielded following information:x1 =
18.7, s21 = 35.1, x2 = 16.4, s22 = 58.2 (i) If your objective is to show that µ1 is
larger than µ2 , state the null and alternative hypotheses. (ii) Is the test in part
(i) one- or a two- tailed test? (iii) Give the test statistic that you would use
for the test in part (i). (iv) State the underline assumptions (v) Calculate the
value of the test statistic. (vi) Draw your conclusion at α = 0.10.
38. The weather bureau measured the following ozone levels (in some units) at 5
locations in a particular village before and after a cool front moved through the
area.
location
before
After
1
2
3
4
5
0.13 0.15 0.09 0.14 0.10
0.10 0.12 0.07 0.10 0.08
Is there a significance drop in ozone level? Decide at the level of significance 0.05. State clearly your hypotheses, test statistic and critical value on
the basis of which you support your decision.
39. Independent random samples from two normal populations with common variance gave the following results:
ni
sample size(ni ) x¯
¯i )2
j=1 (xij − x
i=1
15
35.2 35
i=2
12
34.0 25
P
Can you conclude that the mean of the first population is greater than
the second at α = 0.05?
40. (a) Suppose we want to compare fasting serum cholesterol levels among Asian
immigrants with general female population of United states. It is assumed that
cholesterol levels in women aged 21-40 in USA are approximately normally distributed with mean 170 mg/dL and standard deviation 40mg/dL. It is beleived
that the cholesterol levels among Asians (of same age group) are also normally
distributed with mean 185mg/dL and the same standard deviation as of USA
women. One hundred blood samples of female Asians immigrants gave a sample
x¯ = 181.25 mg/dL . Test the hypothesis that the mean cholesterol level of Asian
immigrants is higher than that of USA female population at α = 0.05. State
the critical region of the test procedure. Compute the power of the test when
µ1 = 185mg/dL.
(b) Suppose it is desired to conduct a test at 1% level of significance with power
of 90% for the two hypotheses that you have considered in part (a). How large
should you take the sample size for the test procedure?
41. Let a random sample of size 16 from N(µ, σ 2 ) yield x = 4.7 and s2 = 5.76. Find
the 90% confidence interval for µ.
5
42. Suppose that a random sample of eight observations are taken from normal
distribution with both the the mean (µ) and variance (σ 2 ) are unknown and
suppose that the observed values are 3.1, 3.5, 2.6, 3.4, 3.8, 3.0, 2.9, and 2.2.
Find the shortest confidence interval for µ with confidence coefficient 0.90.
43. A random sample of 30 items is taken from a normal distribution with unknown
P
mean µ and s.d. σ = 2.5 Given that 30
i=1 xi = 77, calculate the 90% confidence
interval for the population mean.
44. Suppose that a random sample of size n is taken from a normal distribution
with unknown mean µ and s.d. σ = 5. Calculate the the minimum sample size
so that one can be 95% confident the interval [x − 1, x + 1] contains the true
value of µ.
45. A random sample of 15 items is taken from a N(µx , σx2 ) population that yielded
P15 2
P15
i=1 xi = 222. Another random sample of size 11Pis taken
i=1 xi = 53 and
from a N(µy , σy2 ) population independent of the first sample such that 11
i=1 yi =
P11 2
77 and i=1 yi = 560. Obtain a 95% confidence interval for (µx − µy ) by
assuming the true but unknown variances are equal.
46. If 8.6, 7.9, 8.3, 6.4, 8.4, 9.8, 7.2, 7.8, 7.5 are the observed values of a random sample of size 9 from N (µ, σ 2 ). Construct a 95% confidence interval for σ 2
47. Discuss the method of finding 100(1−α)% confidence interval for the differences
of means of two normal population when variances are unknown.
6