GATE DA & AI Engineering Probability and Statistics Flashcards

24 sample cards from the Probability and Statistics deck

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

1. What distinguishes a discrete random variable from a continuous one?

   A discrete random variable takes countable values described by a probability mass function (PMF); a continuous one takes values over an interval described by a probability density function (PDF).

2. What two properties must a probability mass function $p(x)$ satisfy?

   $$p(x) \geq 0 \quad \text{for all } x, \qquad \sum_x p(x) = 1$$

3. Give the PMF, mean, and variance of a discrete uniform distribution over $\{1, 2, \ldots, n\}$.

   $P(X=k)=\frac{1}{n}$; mean $\mu = \frac{n+1}{2}$; variance $\sigma^{2} = \frac{n^{2}-1}{12}$.

4. Give the PMF, mean, and variance of a Bernoulli($p$) random variable.

   $P(X=1)=p,\ P(X=0)=1-p$; mean $= p$; variance $= p(1-p)$.

5. Give the PMF, mean, and variance of a Binomial($n,p$) random variable.

   $$P(X=k) = \binom{n}{k} p^{k}(1-p)^{n-k};\quad \mu = np;\quad \sigma^{2} = np(1-p)$$

6. What two properties must a probability density function $f(x)$ satisfy?

   $$f(x) \geq 0 \quad \text{for all } x, \qquad \int_{-\infty}^{\infty} f(x)\,dx = 1$$

7. How is the probability of an interval found from a continuous PDF $f(x)$?

   $$P(a \leq X \leq b) = \int_{a}^{b} f(x)\,dx$$

8. Give the PDF, mean, and variance of a continuous uniform distribution on $[a,b]$.

   $f(x)=\frac{1}{b-a}$ for $a\leq x\leq b$; mean $=\frac{a+b}{2}$; variance $=\frac{(b-a)^{2}}{12}$.

9. Give the PDF, mean, and variance of an Exponential($\lambda$) distribution.

   $f(x)=\lambda e^{-\lambda x}$ for $x\geq 0$; mean $=\frac{1}{\lambda}$; variance $=\frac{1}{\lambda^{2}}$.

10. What is the memoryless property of the exponential distribution?

    $$P(X > s + t \mid X > s) = P(X > t)$$ The exponential (and geometric) distribution is the only continuous distribution with this property.

11. Give the PMF, mean, and variance of a Poisson($\lambda$) distribution.

    $$P(X=k) = \frac{\lambda^{k} e^{-\lambda}}{k!};\quad \mu = \lambda;\quad \sigma^{2} = \lambda$$

12. Give the PDF of a Normal distribution $N(\mu, \sigma^{2})$.

    $$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}$$

13. What is the standard normal distribution and how is a normal variable standardized?

    The standard normal $N(0,1)$ has mean $0$ and variance $1$. Any $X \sim N(\mu,\sigma^{2})$ is standardized via $$Z = \frac{X - \mu}{\sigma}$$

14. What is the t-distribution and how does it compare to the standard normal?

    The Student's t-distribution is bell-shaped and symmetric about 0 but has heavier tails. As degrees of freedom $\nu \to \infty$ it approaches the standard normal. It is used for small-sample inference about a mean with unknown variance.

15. How is the chi-squared distribution constructed, and what are its mean and variance?

    If $Z_1,\ldots,Z_k$ are independent standard normals, then $\sum_{i=1}^{k} Z_i^{2} \sim \chi^{2}_k$. Mean $= k$; variance $= 2k$.

16. Define the cumulative distribution function (CDF) of a random variable $X$.

    $$F(x) = P(X \leq x)$$ It is non-decreasing, right-continuous, with $F(-\infty)=0$ and $F(\infty)=1$.

17. How are the PDF and CDF of a continuous random variable related?

    $$F(x) = \int_{-\infty}^{x} f(t)\,dt, \qquad f(x) = \frac{dF(x)}{dx}$$

18. How is the CDF used to compute $P(a < X \leq b)$?

    $$P(a < X \leq b) = F(b) - F(a)$$

19. Define the conditional PDF of $X$ given $Y=y$.

    $$f_{X\mid Y}(x \mid y) = \frac{f_{X,Y}(x,y)}{f_Y(y)}, \qquad f_Y(y) > 0$$

20. State the Central Limit Theorem.

    For i.i.d. random variables with mean $\mu$ and finite variance $\sigma^{2}$, the sample mean's standardized sum converges in distribution to standard normal: $$\frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} \xrightarrow{d} N(0,1)$$ as $n \to \infty$.

21. What are the mean and standard deviation (standard error) of the sample mean $\bar{X}_n$?

    $$E[\bar{X}_n] = \mu, \qquad \operatorname{SD}(\bar{X}_n) = \frac{\sigma}{\sqrt{n}}$$

22. What is the linearity of expectation property?

    $$E[aX + bY + c] = a\,E[X] + b\,E[Y] + c$$ It holds regardless of whether $X$ and $Y$ are independent.

23. How does variance behave under scaling and for a sum of independent variables?

    $\operatorname{Var}(aX + b) = a^{2}\operatorname{Var}(X)$. For independent $X,Y$: $\operatorname{Var}(X+Y) = \operatorname{Var}(X) + \operatorname{Var}(Y)$.

24. For a normal distribution, what fraction of data lies within 1, 2, and 3 standard deviations (empirical rule)?

    Approximately $68\%$ within $\mu \pm \sigma$, $95\%$ within $\mu \pm 2\sigma$, and $99.7\%$ within $\mu \pm 3\sigma$.

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