Random variables formulas

Random variables, expectation and variance, including conditional expectation. 26 formulas for GATE DA Probability and Statistics, each with when to use it. Each is explained in the topic notes.

All 26 Random variables formulas

PMF CONDITIONS (DISCRETE)

p_{X} (x) = P (X = x) \geq 0, x \sum p_{X} (x) = 1

Use the second condition to find an unknown constant in a given PMF.

PDF CONDITIONS (CONTINUOUS)

f_{X} (x) \geq 0, \int_{- \infty}^{\infty} f_{X} (x) d x = 1, P (a \leq X \leq b) = \int_{a}^{b} f_{X} (x) d x

A density can exceed 1 at a point; only its integral is a probability.

CDF

F_{X} (x) = P (X \leq x) = {\sum_{t \leq x} p_{X} (t) \int_{- \infty}^{x} f_{X} (t) d t discrete continuous

The CDF is non-decreasing, right-continuous, with limits 0 at $- \infty$ and 1 at $+ \infty$ . For any random variable, $P (a < X \leq b) = F_{X} (b) - F_{X} (a)$ .

RECOVERING PMF AND PDF FROM THE CDF

f_{X} (x) = \frac{d F _{X} ( x )}{d x}, p_{X} (x) = F_{X} (x) - F_{X} (x^{-})

For a continuous variable differentiate the CDF where it is differentiable; for a discrete variable the PMF is the size of the jump at $x$ .

EXPECTATION

E [X] = x \sum x p_{X} (x) or E [X] = \int_{- \infty}^{\infty} x f_{X} (x) d x

Defined when the sum or integral converges absolutely.

EXPECTATION OF A FUNCTION (LOTUS)

E [g (X)] = x \sum g (x) p_{X} (x) or \int_{- \infty}^{\infty} g (x) f_{X} (x) d x

No need to find the distribution of $g (X)$ . In general $E [g (X)] \neq = g (E [X])$ ; they are equal when $g$ is linear, $g (x) = a x + b$ .

LINEARITY OF EXPECTATION

E [a X + bY + c] = a E [X] + b E [Y] + c, E [i = 1 \sum n X_{i}] = i = 1 \sum n E [X_{i}]

True whenever the expectations exist, with or without independence.

VARIANCE

Var (X) = E [(X - E [X])^{2}] = E [X^{2}] - (E [X])^{2}

Standard deviation $SD = Var (X)$ . Since $Var (X) \geq 0$ , $E [X^{2}] \geq (E [X])^{2}$ , and $Var (X) = 0$ only when $X$ is constant with probability 1. Convention: $Var (X)$ here is the population (distribution) variance. For a list of $n$ data values, the population variance divides by $n$ and the sample variance divides by $n - 1$ .

VARIANCE UNDER A LINEAR CHANGE

Var (a X + b) = a^{2} Var (X)

A shift $b$ does not change the variance; a scale $a$ multiplies it by $a^{2}$ (the sign of $a$ is lost).

INDICATOR VARIABLES

I_{A} = {10 A occurs otherwise, E [I_{A}] = P (A), Var (I_{A}) = P (A) (1 - P (A))

To find the expected number of events $A_{1}, \dots, A_{n}$ that occur, write the count as $\sum I_{A_{i}}$ ; by linearity its mean is $\sum P (A_{i})$ , with no independence needed.

MARGINALS

p_{X} (x) = y \sum p_{X, Y} (x, y), f_{X} (x) = \int_{- \infty}^{\infty} f_{X, Y} (x, y) d y

Sum (or integrate) out the other variable over its whole range. Mind the limits when the support is not a rectangle.

INDEPENDENCE

p_{X, Y} (x, y) = p_{X} (x) p_{Y} (y) for all x, y (f_{X, Y} (x, y) = f_{X} (x) f_{Y} (y))

For a PMF the product rule must hold at every pair $(x, y)$ , so one failing pair shows dependence. For a PDF it must hold everywhere apart from sets of zero probability. If the set where the joint PMF or PDF is positive is not a product set $A \times B$ (for example the triangle $0 < x < y < 1$ ), then $X$ and $Y$ are dependent.

COVARIANCE

Cov (X, Y) = E [(X - E [X]) (Y - E [Y])] = E [X Y] - E [X] E [Y]

$Cov (X, X) = Var (X)$ ; $Cov (a X + b, c Y + d) = a c Cov (X, Y)$ .

CORRELATION COEFFICIENT

ρ_{X Y} = \frac{Cov ( X , Y )}{Var ( X ) Var ( Y )}, - 1 \leq ρ_{X Y} \leq 1

Defined when both variances are positive. $∣ ρ ∣ = 1$ exactly when $Y = a X + b$ with probability 1 for some constants $a \neq = 0$ and $b$ ; then $ρ = 1$ if $a > 0$ and $ρ = - 1$ if $a < 0$ .

VARIANCE OF A SUM

Var (a X + bY) = a^{2} Var (X) + b^{2} Var (Y) + 2 ab Cov (X, Y)

For uncorrelated (in particular independent) variables the covariance term vanishes, so $Var (X \pm Y) = Var (X) + Var (Y)$ . Note the plus sign, even for the difference.

VARIANCE OF A SUM OF N VARIABLES

Var (i = 1 \sum n X_{i}) = i = 1 \sum n Var (X_{i}) + 2 i < j \sum Cov (X_{i}, X_{j})

For independent identically distributed $X_{i}$ with variance $σ^{2}$ , the sample mean has $Var (\overset{ˉ}{X}) = σ^{2} / n$ .

PRODUCT OF INDEPENDENT VARIABLES

X, Y independent \Rightarrow E [X Y] = E [X] E [Y], Cov (X, Y) = 0

The converse is false: zero covariance (uncorrelated) does not imply independence.

CONDITIONAL PMF AND PDF

p_{X ∣ Y} (x ∣ y) = \frac{p _{X, Y} ( x , y )}{p _{Y} ( y )}, f_{X ∣ Y} (x ∣ y) = \frac{f _{X, Y} ( x , y )}{f _{Y} ( y )}

Requires $p_{Y} (y) > 0$ (or $f_{Y} (y) > 0$ ). If $X, Y$ are independent the conditional equals the marginal.

CONDITIONAL EXPECTATION

E [X ∣ Y = y] = x \sum x p_{X ∣ Y} (x ∣ y) or \int x f_{X ∣ Y} (x ∣ y) d x

For each fixed $y$ this is a number; $E [X ∣ Y]$ is the random variable obtained by replacing $y$ with $Y$ .

CONDITIONAL VARIANCE

Var (X ∣ Y) = E [(X - E [X ∣ Y])^{2} ∣ Y] = E [X^{2} ∣ Y] - (E [X ∣ Y])^{2}

Also a random variable, a function of $Y$ .

LAW OF TOTAL EXPECTATION

E [X] = E [E [X ∣ Y]] = y \sum E [X ∣ Y = y] P (Y = y)

Replace the sum by an integral against $f_{Y}$ when $Y$ is continuous. For events $A_{1}, A_{2}, \dots$ that partition the sample space, $E [X] = \sum_{i} E [X ∣ A_{i}] P (A_{i})$ .

LAW OF TOTAL VARIANCE

Var (X) = E [Var (X ∣ Y)] + Var (E [X ∣ Y])

Average within-group variance plus variance of the group means.

USEFUL PROPERTIES

E [a X + b Z ∣ Y] = a E [X ∣ Y] + b E [Z ∣ Y], E [g (Y) X ∣ Y] = g (Y) E [X ∣ Y], E [X ∣ Y] = E [X] if X, Y independent

Once $Y$ is given, any function of $Y$ behaves like a constant. The last property does not work in reverse: $E [X ∣ Y] = E [X]$ does not imply independence.

RANDOM SUM

S = i = 1 \sum N X_{i} : E [S] = E [N] μ, Var (S) = E [N] σ^{2} + Var (N) μ^{2}

Here $X_{1}, X_{2}, \dots$ are independent and identically distributed with mean $μ$ and variance $σ^{2}$ , and $N$ is a non-negative integer random variable independent of the $X_{i}$ . It follows from the two total laws with $E [S ∣ N] = N μ$ and $Var (S ∣ N) = N σ^{2}$ .

MARKOV'S INEQUALITY

P (X \geq a) \leq \frac{E [ X ]}{a}

For $X \geq 0$ and $a > 0$ . Needs only the mean.

CHEBYSHEV'S INEQUALITY

P (∣ X - μ ∣ \geq k σ) \leq \frac{1}{k ^{2}}, equivalently P (∣ X - μ ∣ \geq t) \leq \frac{σ ^{2}}{t ^{2}}

Holds for any distribution with mean $μ$ and finite variance $σ^{2}$ ; $k, t > 0$ . Complementary form: $P (∣ X - μ ∣ < k σ) \geq 1 - \frac{1}{k ^{2}}$ .

Where this was asked: 2024 (2), 2025 (4), 2026 (2). See the 8 previous-year questions on Random variables. This shows what was asked, not what will be asked.

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Random variables: the notes and 8 solved previous-year questions