Joint, Conditional and Transformed Random Variables Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Joint, Conditional and Transformed Random Variables short notes for GATE DA: 4 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Summary: Conditional Expectation Master Sheet

Master Sheet

Concept	Formula
Conditional Density	$f_{Y ∣ X} (y ∣ x) = \frac{f _{X, Y} ( x , y )}{f _{X} ( x )}$
Conditional Expectation (Scalar)	$E [Y ∣ X = x] = \int y \cdot f_{Y ∣ X} (y ∣ x) d y$
Conditional Expectation (RV)	$E [Y ∣ X] = g (X)$
Adam's Rule	$E [E [Y ∣ X]] = E [Y]$
Eve's Rule	$V a r (Y) = E [V a r (Y ∣ X)] + V a r (E [Y ∣ X])$

Exam Execution Checklist

Check the support region inequalities to set correct integration limits.
Marginalize correctly before dividing to find the conditional density.
If asked for $E [Y]$ , check if $E [Y ∣ X]$ is easier to compute first (use Adam's Rule).
Distinguish clearly between $E [Y ∣ X = x]$ (a number) and $E [Y ∣ X]$ (a random variable).

Summary: Correlation and Transformations Master Sheet

Master Sheet

Concept	Formula
Covariance	$Cov (X, Y) = E [X Y] - E [X] E [Y]$
Correlation	$ρ_{X, Y} = \frac{Cov ( X , Y )}{σ _{X} σ _{Y}}$
Linear Transform	$Corr (a X + b, c Y + d) = sign (a c) ρ_{X, Y}$
PDF Method	$f_{Y} (y) = f_{X} (g^{- 1} (y)) \frac{d}{d y} g^{- 1} (y)$

Exam Execution Checklist

Correlation via Conditioning: Use $Cov (X, Y) = E [X \cdot E [Y ∣ X]] - E [X] E [Y]$ to bypass complex joint integrations.
Transformations: Check if $g (X)$ is monotonic. If yes, use the PDF formula. If no, use the CDF method or split into branches.
Support Mapping: Never forget to calculate the new boundaries for $Y$ based on the boundaries of $X$ .
Independence Trap: Zero correlation $\neq =$ independence (unless jointly normal).

Exam Readiness: Discrete Transformation Checklist

Identify the Target Type:

Is $Y$ discrete? (Look for $⌊ \cdot ⌋$ , $⌈ \cdot ⌉$ , rounding, or $I (\cdot)$ ).

Translate the Event:

Write the event ${Y = y}$ strictly in terms of inequalities for $X$ .

Determine the Integration Bounds:

Find the exact interval(s) $[a, b)$ of $X$ that satisfy the event.

Calculate the Probability:

Integrate $f_{X} (x)$ over $[a, b)$ , or evaluate $F_{X} (b) - F_{X} (a)$ .

Verify the PMF:

Ensure $\sum p_{Y} (y) = 1$ to catch algebraic errors.

Exam Readiness: Chi-Square Checklist

Standardize:

Ensure the base variable inside the square is $N (0, 1)$ . If it is $N (μ, σ^{2})$ , subtract $μ$ and divide by $σ$ first.

Count Degrees of Freedom ( $k$ ):

Count the number of independent squared standard normal variables being summed.

Apply Moments:

Mean $= k$ , Variance $= 2 k$ .

Use Additivity:

If adding independent chi-square variables, simply add their degrees of freedom ( $k_{t o t a l} = k_{1} + k_{2} + \dots$ ).

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Joint, Conditional and Transformed Random Variables Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Joint, Conditional and Transformed Random Variables short notes for GATE DA: 4 study cards covering concepts, formulas, shortcuts and exam traps, plus solved

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