Probability, Sampling and Statistical Inference - Statistics, Econometrics and Mathematical Economics - Economics Previous Year Questions

Let A and B be two events with probabilities \(P(A) = \frac{3}{4}\) and \(P(B) = \frac{1}{3}\), then which of the following options is true?

If \(S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2\) is an unbiased and consistent estimator of the population variance, then one can conclude that \(S = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2}\) is a(an) __________ estimator of the population standard deviation.

Consider that a sample of size 3 is randomly drawn from a population that takes only two values, equally likely: – 1 and 1. Let \(z = \max (x_1, x_2, x_3)\) where \(x_1, x_2, x_3\) are the sample observations. The expected value of \(z\), \(E(z)\) is __________ (round off to two decimal places).

Let \( f(x) \) be the probability distribution function of a random variable \( X \), where
\( f(x) = (5x^4/64) \) if \( -2 < x < 2 \)
\( \qquad = 0 \) otherwise
Let \( |\cdot| \) denote the modulus function. Then, \( P(|X| < 1) \) and \( P(X^2 < 3) \) are given by (respectively):

A producer manufactures batteries by using techniques I and II. The capacities (in ampere hours) of 12 randomly selected batteries manufactured by using technique I are: 140, 132, 136, 142, 138, 150, 154, 150, 152, 136, 144 and 142. Moreover, the capacities (in ampere hours) of 14 randomly selected batteries manufactured by using technique II are: 144, 134, 132, 130, 136, 146, 140, 128, 131, 128, 150, 137, 130 and 135.
Suppose that battery capacities manufactured by using techniques I and II are normally distributed, with unknown means, \( \mu_I \) and \( \mu_{II} \) respectively, and an unknown common variance \( \sigma^2 \). Consider the null hypothesis \( H_0: (\mu_I - \mu_{II}) = \gamma \) ampere hours.
For which of the following values of \( \gamma \) should the null hypothesis \( H_0 \) be accepted, against the alternative hypothesis \( H_1: (\mu_I - \mu_{II}) \neq \gamma \) ampere hours, at 10% level of significance?
Note that if Z is a standard normal variate, then: \( P(Z \leq 1.282) = 0.900 \), \( P(Z \leq 1.645) = 0.950 \), \( P(Z \leq 1.960) = 0.975 \) and \( P(Z \leq 2.326) = 0.990 \)
Further, if a random variable T follows the Student's \( t \)-distribution, with degrees of freedom \( r \), then the \( \alpha \)-percentile values \( t_{\alpha,r} \) for various values of \( \alpha \) and \( r \) are given by (where \( P(T \leq t_{\alpha,r}) = \alpha \)):
\( t_{0.90,24} = 1.318 \), \( t_{0.90,25} = 1.316 \), \( t_{0.90,26} = 1.315 \), \( t_{0.95,24} = 1.711 \), \( t_{0.95,25} = 1.708 \), \( t_{0.95,26} = 1.706 \), \( t_{0.975,24} = 2.064 \), \( t_{0.975,25} = 2.060 \), \( t_{0.975,26} = 2.056 \).

Let \(X\) be a random variable with the probability density function \(f(x)\) such that \[ f(x) = \begin{cases} \frac{1}{2\sqrt{3}} & \text{if } -\sqrt{3} < x < \sqrt{3} \\ 0 & \text{otherwise} \end{cases} \] Then, the variance of \(X\) is ________ (in integer).

Let \(x_1, x_2, \cdots, x_n\) be an independently, and identically distributed (iid) random sample drawn from a population that follows the Normal Distribution \(N(\mu, \sigma^2)\), where both the mean (\(\mu\)) and variance (\(\sigma^2\)) are unknown. Let \(\bar{x}\) be the sample mean. The maximum likelihood estimator (MLE) of the variance (\(\hat{\sigma}^2_{MLE}\)) is/are then characterized by

For the following function \(f(x)\) to be a probability density function, the value of \(c\) will be __________ (rounded off to two decimal places). \[ f(x) = \begin{cases} \frac{c}{\sqrt{x}}, & 0 < x < 4 \text{ and } c > 0 \\ 0, & \text{otherwise} \end{cases} \]

A six-face fair die is rolled once, with \(X\) being the number that appeared on the uppermost surface. Then the variance of \(X\) is __________ (rounded off to three decimal places).

If \(X\) and \(Y\) are two random variables with the joint probability density function \[ f(x, y) = \begin{cases} \frac{2}{3}(x + 2y); & \text{for } 0 < x, y < 1 \\ 0; & \text{otherwise} \end{cases} \] then \(E\left[X|Y = \frac{1}{2}\right]\) will be

If a discrete random variable \(X\) follows the uniform distribution and assumes only the values 8, 9, 11, 15, 18, and 20, then \(P(|X - 14| < 5)\) is

Assume the following probabilities for two events, \(A\) and \(B\): \(P(A) = 0.50\), \(P(B) = 0.70\), and \(P(A \cup B) = 0.85\). Then we can conclude that

If \( X \) is a continuous random variable whose probability density function is given by \( f_X(x) = \begin{cases} \frac{1}{x^2} & \text{for } 1 < x < \infty \\ 0 & \text{elsewhere} \end{cases} \) Then the median of \( X \) is ______ (in integer)

If \( X \) is a continuous random variable whose probability density function is given by \( f_X(x) = \begin{cases} cx^3 + 0.25 & \text{for } 0 \leq x \leq 1, c \in \mathbb{R} \\ 0 & \text{elsewhere} \end{cases} \) Then the value of \( c \) is ______ (in integer)

Let \(X_1, X_2, X_3, ...\), be independent and identically distributed random variables with \(E[X_1] = \mu\).
Let \(N\) be a positive integer valued random variable with \(E[N] = n\).
If \(S_N = X_1 + X_2 + \cdots + X_N\), then \(E[S_N] =\)

A coin has a true probability \(\mu\) of turning up Heads. This coin is tossed 100 times and shows up Heads 60 times. The following hypothesis is tested: \[ H_0: \mu = 0.5 \text{ (Null Hypothesis)} \] \[ H_1: \mu > 0.5 \text{ (Alternative Hypothesis)} \] Using the Central Limit Theorem, the p-value of the above test is __________ (round off to three decimal places) (Hint: If \(Z\) is a random variable that follows a standard normal distribution, then \(P(Z \leq 2) = 0.977\))

Suppose, a random variable $X$ follows a Normal distribution with mean $\mu$, and variance $\sigma^2$. The variance of the distribution is unknown. For testing $\mu = 10$ against the alternative $\mu > 10$, the distribution of the test statistic is