76
A 95% confidence interval for λ, when a large sample is taken from a Poisson population with parameter λ is
Answer
C. `bar x+- 1.96sqrt((bar x)/(n)) `
77
The minimum Chi-squared estimators are not necessarily
Answer
A. Unbiased
78
Which one of the following statements is true?
Answer
A. Even if the UMP test does not exist, a UMPU test may exist
79
In paired `t` test the two random variables should be
Answer
C. Both paired and correlated
80
With usual notations, the criterion for acceptance in SPRT is
Answer
C. `lambda_m <=(beta)/((1-alpha)) `
81
The Poisson process with parameter λ is a renewal counting process for which the unit lifetimes have ................ distribution with common parameter λ.
Answer
B. Exponential
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
Answer
D. 1
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) = .....`
Answer
A. `(1)/(mu) `
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 ......................
Answer
C. Multinomial with parameters `n` and `(1)/(k)`
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
Answer
B. Circular Normal
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
Answer
B. `sum_(i=1)^k r_i = p`
87
The relationship between partial correlation coefficients `r_{ij.k},` multiple correlation
coefficients `R_{i.jk}`and simple correlation coefficients `r_{ij}` is
coefficients `R_{i.jk}`and simple correlation coefficients `r_{ij}` is
Answer
B. `R^2_1.23 = 1 - (1- r^2_12) (1- r^2_13.2)`
88
Hotelling's `T^2` statistic and Mahalnobis `D^2` statistic are connected by the relationship
Answer
D. `D^2 = ((N_1 + N_2))/((N_1 N_2)) T^2`
89
In principal component analysis the variances of the Principal Components are the .................... of the covariance matrix.
Answer
B. eigen values
90
For discriminating between two populations R.A. Fisher suggested the linear discriminant function `X'l` for which
Answer
A. ` ("(mean difference)"^2)/("variance")`
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
Answer
D. `(lambda (t+2)^2)/(e^(lambda (t+2)))`
92
The basic elements of statistical decision theory is
Answer
D. all these
93
When there is no censoring for the life length `T`, the general formula of a survival function is
Answer
A. `hat {S (t)} = (" # of individuals with" T >= t)/("total sample size")`
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) =`
Answer
C. `h_0 (t) e sum_(i=1)^p beta_i X_i`
95
When an inspection lot contains no defectives the OC function `L (p)` is
Answer
A. `L (p) = 1`
96
In a Time series data, the two main components which cause lack of stationarity are
Answer
D. Trend and seasonal variations
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
Answer
A. ` | phi | <= 1 and | theta | < 1` with `phi != theta`
98
In a Linear programming Problem with `n + m` variables and `m` constraints the number of basic solutions is
Answer
A. ` ((n+m),(m))`
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
Answer
C. `eta_p = 1//bx`
100
The Engel's curves for constant prices and those for constant incomes are respectively
Answer
B. Convex and Concave
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