Lebesgue Measure and Integration - Real Analysis - Mathematics Previous Year Questions

1

2008 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2008

Let \(X_1, X_2, \ldots, X_{10}\) be a random sample from \(N(80, 5^2)\) distribution. Define \(S = \sum_{i=1}^{10} U_i\) and \(T = \sum_{i=1}^{10} \left( U_i - \frac{S}{10} \right)^2\), where \(U_i = \frac{X_i - 80}{5}\), \(i = 1, 2, \ldots, 10\). Then the value of \(E(ST)\) is equal to

A

0

B

1

C

10

D

\(\frac{80}{3}\)

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2

2008 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2008

For each \(n \in \mathbb{N}\), let \(f_n: [0, 1] \to \mathbb{R}\) be a measurable function such that \(|f_n(t)| \le \frac{1}{\sqrt{t}}\) for all \(t \in (0, 1]\). Let \(f: [0, 1] \to \mathbb{R}\) be defined by \(f(t) = 1\) if \(t\) is irrational and \(f(t) = -1\) if \(t\) is rational. Assume that \(f_n(t) \to f(t)\) as \(n \to \infty\) for all \(t \in [0, 1]\). Then

A

\(f\) is not measurable

B

\(f\) is measurable and \(\int_{[0, 1]} f_n d\mu \to 1\) as \(n \to \infty\)

C

\(f\) is measurable and \(\int_{[0, 1]} f_n d\mu \to 0\) as \(n \to \infty\)

D

\(f\) is measurable and \(\int_{[0, 1]} f_n d\mu \to -1\) as \(n \to \infty\)

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3

2008 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2008

Consider the function \[ f(x) = \begin{cases} k(x - [x]), & 0 \le x < 2 \\ 0, & \text{otherwise}, \end{cases} \] where \([x]\) is the integral part of \(x\). The value of \(k\) for which the above function is a probability density function of some random variable is

A

\(\frac{1}{4}\)

B

\(\frac{1}{2}\)

C

1

D

2

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4

2008 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2008

In an examination there are 80 questions each having four choices. Exactly one of these four choices is correct and the other three are wrong. A student is awarded 1 mark for each correct answer, and \(-0.25\) for each wrong answer. If a student ticks the answer of each question randomly, then the expected value of his/her total marks in the examination is

A

-15

B

0

C

5

D

20

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5

2008 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2008

If \(I = \int_{-1}^{1} y(x) x^{n-3} dx\) and \(J = \int_{-1}^{1} y(x) x^n dx\), then

A

\(I \neq 0, J \neq 0\)

B

\(I \neq 0, J = 0\)

C

\(I = 0, J \neq 0\)

D

\(I = 0, J = 0\)

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6

2008 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2008

Let a random variable \(X\) follow the exponential distribution with mean 2. Define \(Y = [X - 2 | X > 2]\). The value of \(P(Y \geq t)\) is

A

\(e^{-t/2}\)

B

\(e^{-2t}\)

C

\(\frac{1}{2} e^{-t/2}\)

D

\(\frac{1}{2} e^{-t}\)

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7

2008 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2008

The value of \(E(Y)\) is equal to

A

\(\frac{1}{4}\)

B

\(\frac{1}{2}\)

C

1

D

2

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8

2009 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2009

A function f : ℝ → ℝ need NOT be Lebesgue measurable if

A

f is monotone

B

{x ∈ ℝ : f(x) ≥ α} is measurable for each α ∈ ℚ

C

{x ∈ ℝ : f(x) = α} is measurable for each α ∈ ℝ

D

For each open set G in ℝ, f^–1(G) is measurable

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9

2009 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2009

Let \(F\), \(G\) and \(H\) be pairwise independent events such that \(P(F) = P(G) = P(H) = \frac{1}{3}\) and \(F \cap G \cap H = \frac{1}{4}\). Then the probability that at least one event among \(F\), \(G\) and \(H\) occurs is

A

\(\frac{11}{12}\)

B

\(\frac{7}{12}\)

C

\(\frac{5}{12}\)

D

\(\frac{3}{4}\)

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10

2009 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2009

Let \(X\) be a random variable such that \(E(X^2) = E(X) = 1\). Then \(E(X^{100}) =\)

A

0

B

1

C

\(2^{100}\)

D

\(2^{100} + 1\)

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11

2009 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2009

For which of the following distributions, the weak law of large numbers does NOT hold ?

A

Normal

B

Gamma

C

Beta

D

Cauchy

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12

2009 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2009

Let \( X \) be a non-negative integer valued random variable with \( E(X^2) = 3 \) and \( E(X) = 1 \). Then \( \sum_{i=1}^\infty i P(X \ge i) = \)

A

1

B

2

C

3

D

4

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13

2009 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2009

Let \( X \) be a random variable with probability density function \( f \in \{ f_0, f_1 \} \), where \( f_0(x) = \begin{cases} 2x, & \text{if } 0 < x < 1 \\ 0, & \text{otherwise} \end{cases} \) and \( f_1(x) = \begin{cases} 3x^2, & \text{if } 0 < x < 1 \\ 0, & \text{otherwise} \end{cases} \). For testing the null hypothesis \( H_0 : f = f_0 \) against the alternative hypothesis \( H_1 : f = f_1 \) at level of significance \( \alpha = 0.19 \), the power of the most powerful test is

A

0.729

B

0.271

C

0.615

D

0.385

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14

2009 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2009

The variance of the random variable \( X \) is

A

\(\frac{1}{12} \)

B

\(\frac{1}{4} \)

C

\(\frac{7}{12} \)

D

\(\frac{5}{12} \)

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15

2009 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2009

The covariance between the random variables \( X \) and \( Y \) is

A

\(\frac{1}{3} \)

B

\(\frac{1}{4} \)

C

\(\frac{1}{6} \)

D

\(\frac{1}{12} \)

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16

2010 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2010

If \( f: [1, 2] \to \mathbb{R} \) is a non-negative Riemann-integrable function such that \( \int_1^2 \frac{f(x)}{\sqrt{x}} dx = k \int_1^2 f(x) dx = 0 \), then \( k \) belongs to the interval

A

\(\left[0, \frac{1}{3}\right] \)

B

\(\left(\frac{1}{3}, \frac{2}{3}\right] \)

C

\(\left(\frac{2}{3}, 1\right] \)

D

\(\left(1, \frac{4}{3}\right] \)

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17

2010 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2010

Let \( A \) and \( B \) be disjoint subsets of \( \mathbb{R} \) and let \( m^* \) denote the Lebesgue outer measure on \( \mathbb{R} \). Consider the statements:
\( P: m^*(A \cup B) = m^*(A) + m^*(B) \)
\( Q: \) Both \( A \) and \( B \) are Lebesgue measurable
\( R: \) One of \( A \) and \( B \) is Lebesgue measurable
Which one of the following is correct?

A

If \(P \) is true, then \(Q \) is true

B

If \(P \) is NOT true, then \(R \) is true

C

If \(R \) is true, then \(P \) is NOT true

D

If \(R \) is true, then \(P \) is true

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18

2010 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2010

Let \( f: \mathbb{R} \to [0, \infty) \) be a Lebesgue measurable function and \( E \) be a Lebesgue measurable subset of \( \mathbb{R} \) such that \( \int_E f dm = 0 \), where \( m \) is the Lebesgue measure on \( \mathbb{R} \). Then

A

\(m(E) = 0 \)

B

\(\{x \in E : f(x) = 0\} = E \)

C

\(m(\{x \in E : f(x) \neq 0\}) = 0 \)

D

\(m(\{x \in E : f(x) = 0\}) = 0 \)

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19

2010 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2010

The value of \( E(X | Y = 2) \) is

A

4

B

3

C

2

D

1

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20

2011 · Mathematics · Real Analysis · Lebesgue Measure and Integration

Mathematics (MA) 2011

Which of the following functions is a probability density function of a random variable \(X\)?

A

\(f(x) = \begin{cases} x(2-x), & 0 < x < 2 \\ 0, & \text{elsewhere} \end{cases}\)

B

\(f(x) = \begin{cases} x(1-x), & 0 < x < 1 \\ 0, & \text{elsewhere} \end{cases}\)

C

\(f(x) = \begin{cases} 2xe^{-x^2}, & -1 < x < 1 \\ 0, & \text{elsewhere} \end{cases}\)

D

\(f(x) = \begin{cases} 2xe^{-x^2}, & x > 0 \\ 0, & \text{elsewhere} \end{cases}\)

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