81.
The
Poisson process with parameter Î» is a renewal counting process for which the
unit lifetimes have ................ distribution with common parameter Î».

(A)
Poisson

(B)
Exponential

(C)
Uniform

(D)
Geometric

Answer: B

82.
Let
`{X_n, n = 0, 1, 2...}` be a Branching process and the corresponding offspring
distribution has a pgf `P (s)=(2)/(3) +(s+s^2)/(6) ` . Find the probability of
extinction of the process

(A)
0

(B)
0.25

(C)
0.66

(D)
1

Answer: D

83.
Let
`{X_n}` be a renewal process with `mu = E (X_1) <oo ` and if `M (t)` is the
renewal function, then `lim_(t->oo) (M (t))/(t) = .....`

(A)
`(1)/(mu) `

(B)
`mu `

(C)
`(t)/(mu) `

(D)
`(mu)/(t) `

Answer: A

84.
If
`X_i`'s are independent Poisson variates with respective parameters `lambda_i`,
for `i = 1, 2, ...k`, then the conditional distribution of `X_1, X_2,... X_k`
given their sum `sum_(i=1)^k X_i = n` is a ................. distribution with parameters
....................... and ......................

(A)
Binomial with parameters `n` and ` (1)/(k)`

(B)
Binomial with parameters `k` and `(1)/(n)`

(C)
Multinomial with parameters `n` and `(1)/(k)`

(D)
Multinomial with parameters `k` and `(1)/(n)`

Answer: C

85.
If
` (X_1, X_2)` is a Bivariate normal random vector with parameters `(mu_{X1},
mu_{X2}, sigma ^2_X_1,sigma^2_X_2, rho`), when `sigma^2_X_1 = sigma^2_X_2 ` and
`rho = 0` , the density function is called

(A)
Elliptical Normal

(B)
Circular Normal

(C)
Symmetrical Normal

(D)
Uniform Normal

Answer: B

86.
If
the random vector `X` follows Multivariate Normal distribution with mean vector
0 and dispersion matrix `I` and `Q_i = X^' A_i X` are quadratic forms of rank
`r_i` such that `sum_(i=1)^k A_i = I_p` , then a necessary and sufficient condition
for `Q_i`'s to be distributed as independent chi-square random variables with
`r_i` d.f is that

(A)
` sum_(i=1)^k r_i=k`

(B)
`sum_(i=1)^k r_i = p`

(C)
`sum_(i=1)^k r_i=0`

(D)
`sum_(i=1)^k r_i = kp`

Answer: B

87.
The
relationship between partial correlation coefficients `r_{ij.k},` multiple
correlation

coefficients
`R_{i.jk}`and simple correlation coefficients `r_{ij}` is

(A)
`R^2_1.23 = 1+ (1-r^2_12) (1 - r^2_13.2)`

(B)
`R^2_1.23 = 1 - (1- r^2_12) (1- r^2_13.2)`

(C)
`R^2_1.23 = 1 + (1-r^2_12)// (1-r^2_13.2)`

(D)
`R^2_1.23 = 1- (1-r^2_12) // (1-r^2_13.2)`

Answer: B

88.
Hotelling's
`T^2` statistic and Mahalnobis `D^2` statistic are connected by the
relationship

(A)
`D^2 = ((N_1 N_2))/((N_1+N_2)) T^2`

(B)
`D^2 =((N_1 N_2))/((N_1-N_2)) T^2`

(C)
`D^2 = ((N_1-N_2))/((N_1 N_2)) T^2`

(D)
`D^2 = ((N_1 + N_2))/((N_1 N_2)) T^2`

Answer: D

89.
In
principal component analysis the variances of the Principal Components are the ....................
of the covariance matrix.

(A)
diagonal elements

(B)
eigen values

(C)
normalized elements

(D)
non-zero elements

Answer: B

90.
For
discriminating between two populations R.A. Fisher suggested the linear discriminant
function `X'l` for which

(A)
` ("(mean difference)"^2)/("variance")`

(B)
`("mean difference")^2/("A.M.")`

(C)
`("mean difference")/("median")`

(D)
`("variance")/("mean difference")`

Answer: A

91.
Assume
that the time to failure `(T)` for a certain bulb has an exponential
distribution `f ((t)/(lambda))` with parameter `lambda >0` with the prior
pdf `g (lambda)` of `lambda` is an exponential distribution with parameter 2.
Then the posterior pdf of `lambda` given `T = t` is

(A)
`(2)/(t+2)`

(B)
`(lambda)/(e^lambda (t+2))`

(C)
`(lambda e^(lambda (t+2)))/((t+2)^2)`

(D)
`(lambda (t+2)^2)/(e^(lambda (t+2)))`

Answer: D

92.
The
basic elements of statistical decision theory is

(A)
a space Î© `= {ul theta}` of all possible states of nature

(B)
an action space `A = {a}` of all possible courses of action

(C)
a loss function `L (ul theta, a)` giving the incurred loss when action `a` is
taken and the state is ` ul theta`

(D)
all these

Answer: D

93.
When
there is no censoring for the life length `T`, the general formula of a survival
function is

(A)
`hat {S (t)} = (" # of individuals with" T >= t)/("total
sample size")`

(B)
`hat {S (t)} = (" # of individuals with" T <= t)/("total
sample size")`

(C)
`hat {S (t)} = (" # of individuals with" T = t)/("total sample
size")`

(D)
`hat {S (t)} = (" # of individuals with" T = 0)/("total sample
size")`

Answer: A

94.
The
Cox's Proportional Hazard Model (Cox's PH Model) with explanatory variables `
ul X = (X_1, X_2, ... X_p),beta_i` their regression coefficients and `h_0 (t)`
a base line hazard, is `h (t, ul X) =`

(A)
`e^{h_0 (t) sum_(i=1)^p beta_i X_i}`

(B)
`log h_0 (t) + sum_(i=1)^p beta_i X_i`

(C)
`h_0 (t) e sum_(i=1)^p beta_i X_i`

(D)
`e^(h_0 (t)) sum_(i=1)^p beta_i X_i`

Answer: C

95.
When
an inspection lot contains no defectives the OC function `L (p)` is

(A)
`L (p) = 1`

(B)
`L (p) = oo`

(C)
` L (p) = 0`

(D)
None of these

Answer: A

96.
In
a Time series data, the two main components which cause lack of stationarity
are

(A)
Seasonal and irregular variations

(B)
Cyclic and irregular variations

(C)
Trend and cyclic variations

(D)
Trend and seasonal variations

Answer: D

97.
In
the ARMA (1, 1) model `Z_t = ``phi Z_{t-1} + epsilon _t - theta epsilon_{t-1}`
the condition for stationarity and invertilibility are respectively

(A)
` | phi | <= 1 and | theta | < 1` with `phi != theta`

(B)
` | phi | <= 1 and | theta | < 1` with `phi = theta`

(C)
` | phi | > 1 and | theta | > 1` with `phi!= theta`

(D)
` | phi | > 1 and | theta | > 1` with `phi = theta`

Answer: A

98.
In
a Linear programming Problem with `n + m` variables and `m` constraints the
number of basic solutions is

(A)
` ((n+m),(m))`

(B)
`((n),(m))`

(C)
`((m),(n))`

(D)
`((n+m),(n-m))`

Answer: A

99.
If
the demand curve is of the form `p = ae^{-bx}` , where `p` is the price and `x`
is the demand, then the price elasticity of demand is

(A)
`eta_p = bx`

(B)
`eta_p = - bx`

(C)
`eta_p = 1//bx`

(D)
`eta_p = - 1 // bx`

Answer: C

100.
The
Engel's curves for constant prices and those for constant incomes are
respectively

(A)
Concave and Convex

(B)
Convex and Concave

(C)
Both Concave

(D)
Both Convex

Answer: B

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