CSIR NET Mathematical Science UNIT – 4 Flashcards

24 sample cards from the UNIT – 4 deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. Define covariance and the correlation coefficient between $X$ and $Y$.

   $\operatorname{Cov}(X,Y) = E[(X - E X)(Y - E Y)] = E[XY] - E[X]E[Y]$. The correlation coefficient is $$\rho = \frac{\operatorname{Cov}(X,Y)}{\sqrt{\operatorname{Var}(X)\,\operatorname{Var}(Y)}}, \quad -1 \leq \rho \leq 1.$$

2. Define skewness and kurtosis in terms of central moments.

   Skewness $\gamma_1 = \frac{\mu_3}{\mu_2^{3/2}}$ measures asymmetry. Kurtosis $\beta_2 = \frac{\mu_4}{\mu_2^{2}}$ measures tail heaviness/peakedness; excess kurtosis is $\beta_2 - 3$ (zero for the normal distribution).

3. Define the moment generating function (MGF) and state how moments are recovered from it.

   $M_X(t) = E[e^{tX}]$, when it exists in a neighborhood of $0$. Then $E[X^{r}] = M_X^{(r)}(0)$, the $r$-th derivative evaluated at $t=0$. For independent $X,Y$: $M_{X+Y}(t) = M_X(t)M_Y(t)$.

4. Define the characteristic function of a random variable $X$.

   $$\varphi_X(t) = E[e^{itX}] = \int_{-\infty}^{\infty} e^{itx}\,dF_X(x), \quad t \in \mathbb{R}.$$ It always exists (unlike the MGF) since $|e^{itX}| = 1$, and uniquely determines the distribution.

5. List the key properties of the characteristic function $\varphi_X(t)$.

   (1) $\varphi_X(0) = 1$ and $|\varphi_X(t)| \leq 1$. (2) Uniformly continuous on $\mathbb{R}$. (3) $\varphi_X$ uniquely determines the distribution (uniqueness/inversion theorem). (4) For independent $X,Y$: $\varphi_{X+Y}(t) = \varphi_X(t)\varphi_Y(t)$. (5) $\varphi_{aX+b}(t) = e^{itb}\varphi_X(at)$.

6. How are moments obtained from the characteristic function?

   If $E[|X|^{n}] < \infty$, then $\varphi_X$ is $n$ times differentiable and $$E[X^{k}] = \frac{\varphi_X^{(k)}(0)}{i^{k}}, \quad k = 1,\dots,n.$$

7. State the characteristic function of the standard normal $N(0,1)$ and of $N(\mu,\sigma^{2})$.

   For $N(0,1)$: $\varphi(t) = e^{-t^{2}/2}$. For $N(\mu,\sigma^{2})$: $\varphi(t) = \exp\!\left(i\mu t - \tfrac{1}{2}\sigma^{2}t^{2}\right)$.

8. State Markov's inequality and its key requirement.

   For a non-negative random variable $X$ and any $a > 0$, $$P(X \geq a) \leq \frac{E[X]}{a}.$$ It requires $X \geq 0$ and gives an upper bound on tail probability using only the mean.

9. State Tchebyshev's (Chebyshev's) inequality.

   For a random variable $X$ with mean $\mu$ and finite variance $\sigma^{2}$, for any $k > 0$, $$P(|X - \mu| \geq k) \leq \frac{\sigma^{2}}{k^{2}}.$$ Equivalently $P(|X-\mu| \geq k\sigma) \leq \frac{1}{k^{2}}$.

10. How is Chebyshev's inequality derived from Markov's inequality?

    Apply Markov's inequality to the non-negative random variable $(X-\mu)^{2}$: $$P(|X-\mu| \geq k) = P\big((X-\mu)^{2} \geq k^{2}\big) \leq \frac{E[(X-\mu)^{2}]}{k^{2}} = \frac{\sigma^{2}}{k^{2}}.$$

11. State Jensen's inequality and the role of convexity.

    If $g$ is a convex function and $X$ is a random variable with finite mean, then $$g(E[X]) \leq E[g(X)].$$ If $g$ is concave, the inequality reverses: $g(E[X]) \geq E[g(X)]$. Equality holds iff $g$ is linear on the support of $X$ (or $X$ is degenerate).

12. Give two standard consequences of Jensen's inequality.

    Since $x \mapsto x^{2}$ is convex: $E[X^{2}] \geq (E[X])^{2}$, i.e. variance is non-negative. Since $\log$ is concave: $E[\log X] \leq \log E[X]$ (relating geometric and arithmetic means). Also $|E[X]| \leq E|X|$ from convexity of $|\cdot|$.

13. State the Cauchy-Schwarz inequality for random variables.

    For random variables $X,Y$ with finite second moments, $$\big(E[XY]\big)^{2} \leq E[X^{2}]\,E[Y^{2}].$$ This implies the correlation coefficient satisfies $|\rho| \leq 1$.

14. Define convergence in probability of a sequence $X_n$ to $X$.

    $X_n \xrightarrow{P} X$ if for every $\varepsilon > 0$, $$\lim_{n\to\infty} P(|X_n - X| > \varepsilon) = 0.$$

15. Define almost sure convergence of $X_n$ to $X$.

    $X_n \xrightarrow{a.s.} X$ if $$P\left(\lim_{n\to\infty} X_n = X\right) = 1,$$ i.e. $X_n(\omega) \to X(\omega)$ for all $\omega$ outside a set of probability zero.

16. Define convergence in distribution (weak convergence) of $X_n$ to $X$.

    $X_n \xrightarrow{d} X$ if $F_{X_n}(x) \to F_X(x)$ at every continuity point $x$ of $F_X$. Equivalently, $\varphi_{X_n}(t) \to \varphi_X(t)$ for all $t$ (Lévy continuity theorem).

17. Define convergence in $r$-th mean ($L^r$ convergence) of $X_n$ to $X$.

    $X_n \xrightarrow{L^r} X$ if $E[|X_n - X|^{r}] \to 0$ as $n \to \infty$, for $r \geq 1$. The case $r=2$ is convergence in mean square.

18. State the hierarchy of the modes of convergence (which implies which).

    Almost sure convergence $\Rightarrow$ convergence in probability. Convergence in $L^r$ $\Rightarrow$ convergence in probability. Convergence in probability $\Rightarrow$ convergence in distribution. Convergence in distribution is the weakest; none of the reverse implications hold in general (except convergence in distribution to a constant implies convergence in probability).

19. State the Weak Law of Large Numbers (WLLN) for i.i.d. variables.

    If $X_1, X_2, \dots$ are i.i.d. with finite mean $\mu$, then the sample mean $\bar{X}_n = \frac{1}{n}\sum_{i=1}^{n} X_i$ converges to $\mu$ in probability: $$\bar{X}_n \xrightarrow{P} \mu.$$

20. State the Strong Law of Large Numbers (SLLN) for i.i.d. variables.

    If $X_1, X_2, \dots$ are i.i.d. with finite mean $\mu$ (i.e. $E|X_1| < \infty$), then $$\bar{X}_n = \frac{1}{n}\sum_{i=1}^{n} X_i \xrightarrow{a.s.} \mu.$$

21. Contrast the Weak and Strong Laws of Large Numbers.

    WLLN gives convergence in probability ($\bar X_n \xrightarrow{P} \mu$); SLLN gives the stronger almost sure convergence ($\bar X_n \xrightarrow{a.s.} \mu$). SLLN implies WLLN. Kolmogorov's SLLN requires only $E|X_1| < \infty$, while the simplest WLLN proofs use finite variance via Chebyshev.

22. State the classical (Lindeberg-Lévy) Central Limit Theorem for the i.i.d. case.

    If $X_1, X_2, \dots$ are i.i.d. with mean $\mu$ and finite variance $\sigma^{2} > 0$, then $$\frac{\sqrt{n}\,(\bar{X}_n - \mu)}{\sigma} = \frac{\sum_{i=1}^{n} X_i - n\mu}{\sigma\sqrt{n}} \xrightarrow{d} N(0,1).$$

23. How does the Central Limit Theorem differ from the Law of Large Numbers in what they describe?

    The LLN states that $\bar X_n$ converges to the constant $\mu$ (consistency). The CLT describes the fluctuation of $\bar X_n$ around $\mu$ on the scale $\frac{1}{\sqrt{n}}$, asserting that $\sqrt{n}(\bar X_n - \mu)$ is asymptotically normal $N(0,\sigma^{2})$. The CLT gives the rate/shape of convergence.

24. Define a stochastic process with finite or countable state space and what its state space refers to.

    A stochastic process $\{X_t\}$ is a collection of random variables indexed by $t$ (time). Its state space is the set of possible values the $X_t$ take. A finite state space has finitely many states $\{1,2,\dots,N\}$; a countable state space has countably infinitely many states (e.g. $\{0,1,2,\dots\}$), as in a Markov chain on the non-negative integers.

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