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IIT M FOUNDATION FN EXAM FDF1 24 Dec 2023 - 24 Dec 2023

MLF End Term16 questions40 marks8 SA · 5 MCQ · 3 MSQFoundation / forenoon

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Q1MCQ0 marks

Useful Data has been mentioned above.
This data attachment is just for a reference & not for an evaluation.

Q2SA3 marks

Your numeric answer:

Q3SA3 marks

Your numeric answer:

Q4SA3 marks

Your numeric answer:

Q5SA3 marks

Your numeric answer:

Q6SA3 marks

Your numeric answer:

Q7MSQ3 marksselect all that apply

Which of the following functions is/are continuous?

Q8MSQ3 marksselect all that apply

Let A be a n × n positive definite matrix. Then, which among the following are correct?

Q9SA4 marks

Your numeric answer:

Q10MSQ3 marksselect all that apply

Q11MCQ2 marks

Q12MCQ2 marks

Q13MCQ2 marks

Comprehension passage - the questions below it refer to this

Q14MCQ2 marks

Which of the following represents the covariance matrix, C for the given data points?

Q15SA2 marks

What is the value of the projected variance? Enter the answer correct to two decimal places.

Your numeric answer:

Q16SA2 marks

Your numeric answer:

Answer key

Q1 (MCQ, 0 marks): A
Q2 (SA, 3 marks): 9
Q3 (SA, 3 marks): 0.089 to 0.095
Q4 (SA, 3 marks): 0.74 to 0.79
Q5 (SA, 3 marks): 0.24 to 0.28
Q6 (SA, 3 marks): 0.020 to 0.030
Q7 (MSQ, 3 marks): A, B, D
Q8 (MSQ, 3 marks): A, B, D
Q9 (SA, 4 marks): 0.370 to 0.380
Q10 (MSQ, 3 marks): B, C
Q11 (MCQ, 2 marks): C
Q12 (MCQ, 2 marks): B
Q13 (MCQ, 2 marks): B
Q14 (MCQ, 2 marks): C
Q15 (SA, 2 marks): 5.30 to 5.37
Q16 (SA, 2 marks): 10

Concepts this paper tests

Auto-generated summary from the source site. Handy as a checklist; the lessons above explain each idea from scratch.

Here’s a structured summary of core topics, concepts, principles, and equations for the exam, followed by Mermaid knowledge graphs to visualize relationships.

Core Topics & Concepts

1. Probability & Statistics

A. Random Variables (Discrete & Continuous)

Distribution	PMF/PDF	CDF	Mean ( $E[X]$ )	Variance ( $Var(X)$ )
Uniform (Discrete)	$f_X(k) = \frac{1}{n}, \quad k = a, a+1, \ldots, b$	$F_X(x) = \frac{k-a+1}{n}, \quad a \leq x < b$	$\frac{a+b}{2}$	$\frac{n^2-1}{12}$
Bernoulli	$f_X(k) = p^k (1-p)^{1-k}, \quad k \in \{0,1\}$	$F_X(x) = 1-p, \quad 0 \leq x < 1$	$p$	$p(1-p)$
Binomial	$f_X(k) = {}^nC_k p^k (1-p)^{n-k}, \quad k = 0, \ldots, n$	$F_X(x) = \sum_{i=0}^k {}^nC_i p^i (1-p)^{n-i}$	$np$	$np(1-p)$
Geometric	$f_X(k) = (1-p)^{k-1}p, \quad k = 1, 2, \ldots$	$F_X(x) = 1-(1-p)^k, \quad k \leq x < k+1$	$\frac{1}{p}$	$\frac{1-p}{p^2}$
Poisson	$f_X(k) = \frac{e^{-\lambda} \lambda^k}{k!}, \quad k = 0, 1, \ldots$	$F_X(x) = e^{-\lambda} \sum_{i=0}^k \frac{\lambda^i}{i!}$	$\lambda$	$\lambda$
Uniform (Continuous)	$f_X(x) = \frac{1}{b-a}, \quad a \leq x \leq b$	$F_X(x) = \frac{x-a}{b-a}, \quad a \leq x \leq b$	$\frac{a+b}{2}$	$\frac{(b-a)^2}{12}$
Exponential	$f_X(x) = \lambda e^{-\lambda x}, \quad x \geq 0$	$F_X(x) = 1-e^{-\lambda x}, \quad x \geq 0$	$\frac{1}{\lambda}$	$\frac{1}{\lambda^2}$
Normal	$f_X(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$	No closed form	$\mu$	$\sigma^2$
Gamma	$f_X(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}, \quad x > 0$	$F_X(x) = \frac{\Gamma(\alpha, \beta x)}{\Gamma(\alpha)}$	$\frac{\alpha}{\beta}$	$\frac{\alpha}{\beta^2}$
Beta	$f_X(x) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1} (1-x)^{\beta-1}$	No closed form	$\frac{\alpha}{\alpha+\beta}$	$\frac{\alpha \beta}{(\alpha+\beta)^2 (\alpha+\beta+1)}$

B. Theorems & Inequalities

Markov’s Inequality:

P(X \geq c) \leq \frac{E[X]}{c}, \quad X \geq 0

Chebyshev’s Inequality:

P(|X - \mu| \geq k\sigma) \leq \frac{1}{k^2}

Weak Law of Large Numbers (WLLN):

P(|\bar{X} - \mu| > \delta) \leq \frac{\sigma^2}{n\delta^2}

Central Limit Theorem (CLT):

\frac{Y - n\mu}{\sqrt{n}\sigma} \approx \text{Normal}(0,1)

where $Y = X_1 + \ldots + X_n$ .

Maximum Likelihood Estimation (MLE):

\hat{\theta} = \arg \max_\theta \prod_{i=1}^n f_X(x_i; \theta)

For Exponential Distribution:

\hat{\lambda} = \frac{n}{\sum_{i=1}^n U_i}

C. Joint & Conditional Distributions

Joint PDF/PMF: $f_{X,Y}(x,y)$
Marginal PDF/PMF: $f_X(x) = \int f_{X,Y}(x,y) \, dy$
Conditional Expectation:

E[X \mid Y=y] = \int x f_{X \mid Y}(x \mid y) \, dx

2. Linear Algebra

A. Matrix Properties

Positive Definite Matrix:
- $A$ is positive definite if $x^T A x > 0$ for all $x \neq 0$ .
- Properties:
  - All eigenvalues $> 0$ .
  - $A^k$ is positive definite for $k \geq 1$ .
  - $A^{-1}$ is positive definite.
  - Diagonal entries $> 0$ .

Eigenvalues & Eigenvectors:
- $A v = \lambda v$
- If $A$ has $n$ linearly independent eigenvectors, it is diagonalizable.

Nullspace:
- $\text{Null}(A) = \{ x \mid A x = 0 \}$

B. PCA (Principal Component Analysis)

Steps:
1. Center the data: $X_i \leftarrow X_i - \bar{X}$ .
2. Compute covariance matrix $C = \frac{1}{n} \sum (X_i - \bar{X})(X_i - \bar{X})^T$ .
3. Find eigenvalues $\lambda_i$ and eigenvectors $v_i$ of $C$ .
4. Project data onto the top eigenvector(s):

Y = X v_1

Projected Variance:

\text{Var}(Y) = v_1^T C v_1 = \lambda_1

3. Optimization

A. Linear Programming

Standard Form:

\text{Maximize } Z = c^T y

Subject to:

A y \leq b, \quad y \geq 0

Example:

\text{Maximize } Z = 60y_1 + 45y_2

Subject to:

2y_1 + 2y_2 \leq 200, \quad y_1 + 3y_2 \leq 140, \quad y_1, y_2 \geq 0

B. Constrained Optimization

Lagrange Multipliers:

\mathcal{L}(x, y, \lambda) = f(x,y) - \lambda (g(x,y) - c)

Solve:

\nabla \mathcal{L} = 0

Example: Maximize $f(x,y) = x + y$ subject to $x^2 + y^2 = 50$ .

4. Calculus & Continuity

Continuous Functions:
- $f(x)$ is continuous at $x = a$ if:

\lim_{x \to a} f(x) = f(a)

Examples:
- $f(x) = \frac{\sin x}{x}$ (continuous at $x=0$ if defined as $f(0)=1$ ).
- Piecewise functions must match at boundaries.

5. Convexity

Convex Function:

f(\alpha x + (1-\alpha)y) \leq \alpha f(x) + (1-\alpha) f(y), \quad \alpha \in [0,1]

Examples:
- $f(x) = \frac{1}{x}$ (convex for $x > 0$ ).
- $g(x) = 1 + |x|$ (convex everywhere).
Composition Rules:
- If $f$ is convex and $g$ is convex + non-decreasing, then $f \circ g$ is convex.

IIT M FOUNDATION FN EXAM FDF1 24 Dec 2023 - 24 Dec 2023

Answer key

Concepts this paper tests

Core Topics & Concepts

1. Probability & Statistics

A. Random Variables (Discrete & Continuous)

B. Theorems & Inequalities

C. Joint & Conditional Distributions

2. Linear Algebra

A. Matrix Properties

B. PCA (Principal Component Analysis)

3. Optimization

A. Linear Programming

B. Constrained Optimization

4. Calculus & Continuity

5. Convexity

2. Linear Algebra

3. Optimization & PCA

4. Calculus & Convexity