This one took me a minute to even remember where to start.
Recognize that correlation coefficients must form a positive semi-definite correlation matrix. Use the determinant condition to derive the inequality that bounds the possible values of Corr(Y,Z). Then solve for the range and verify with the given values.
Pro tip: Mention that this is a classic application of the Cauchy-Schwarz inequality for random variables, and that the bounds are tight—meaning any value in the range is achievable by some joint distribution.
Write the 3x3 correlation matrix with 1s on the diagonal and the given correlations: Corr(X,Y)=0.8, Corr(X,Z)=0.9, and Corr(Y,Z)=r (unknown).
For a valid correlation matrix, the determinant must be non-negative. Compute the determinant in terms of r and set it ≥ 0.
Expand and simplify the determinant inequality to find the range of r. The determinant is 1 + 2*0.8*0.9*r - 0.8^2 - 0.9^2 - r^2 ≥ 0.
Solve the quadratic inequality: -r^2 + 1.44r + 0.19 ≥ 0, which gives r between approximately -0.115 and 1.0. However, check if the upper bound is exactly 1 or less.
Confirm that the bounds are correct and note that the range is [-0.115, 1.0] (or more precisely, [0.8*0.9 - sqrt((1-0.8^2)(1-0.9^2)), 0.8*0.9 + sqrt((1-0.8^2)(1-0.9^2))] = [0.72 - 0.6*0.43589, 0.72 + 0.6*0.43589] = [0.72 - 0.2615, 0.72 + 0.2615] = [0.4585, 0.9815]). Wait, recalc: sqrt((1-0.64)(1-0.81)) = sqrt(0.36*0.19) = sqrt(0.0684) = 0.2615. So range is [0.72 - 0.2615, 0.72 + 0.2615] = [0.4585, 0.9815]. But earlier determinant gave -0.115? Let's re-evaluate determinant: det = 1 + 2*0.8*0.9*r - 0.8^2 - 0.9^2 - r^2 = 1 + 1.44r - 0.64 - 0.81 - r^2 = -0.45 + 1.44r - r^2. Set ≥0 => r^2 - 1.44r + 0.45 ≤ 0. Roots: (1.44 ± sqrt(2.0736 - 1.8))/2 = (1.44 ± sqrt(0.2736))/2 = (1.44 ± 0.523)/2 => (0.917)/2=0.4585 and (1.963)/2=0.9815. So range is [0.4585, 0.9815]. Correct.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by recognizing that the correlation matrix must be positive semidefinite; for three variables with equal pairwise correlation ρ, the matrix is PSD iff ρ ≥ -1/2. Since ρ = -1/2 is the boundary case, the variables must be perfectly linearly dependent. Construct X and Y as independent Rademacher variables, then set Z = -X - Y. Verify that all pairwise correlations equal -1/2.
Pro tip: Mention the positive semidefinite constraint on correlation matrices and that -1/2 is the minimum possible equal correlation for three variables. This shows you understand the underlying theory, not just a construction.
Recall that any correlation matrix must be positive semidefinite. For three variables with equal pairwise correlation ρ, the eigenvalues are (1+2ρ) and (1-ρ) twice, so PSD requires ρ ≥ -1/2. Thus ρ = -1/2 is the extreme case.
At ρ = -1/2, the correlation matrix is singular, meaning the variables are linearly dependent. This suggests a construction where one variable is a linear combination of the other two.
Let X and Y be independent Rademacher variables (taking values ±1 with equal probability). Define Z = -X - Y. Then compute the correlations: Corr(X,Y)=0, Corr(X,Z)= -1/2, Corr(Y,Z)= -1/2.
Calculate means and variances: E[X]=E[Y]=0, Var(X)=Var(Y)=1, E[Z]=0, Var(Z)=2. Then Cov(X,Z) = E[XZ] = E[-X^2 - XY] = -1, so Corr(X,Z) = -1/√2? Wait, need to check: Actually Cov(X,Z) = E[X(-X-Y)] = -E[X^2] - E[XY] = -1 - 0 = -1. Var(Z)=Var(-X-Y)=Var(X)+Var(Y)=2. So Corr(X,Z) = -1/√2 ≈ -0.707, not -1/2. This construction is incorrect. Let's correct: We need Corr = -1/2. Try Z = aX + bY + c? Actually, we need Cov(X,Z) = -1/2 * sqrt(Var(X)Var(Z)). Let's solve properly.
We need three variables with equal pairwise correlation -1/2. A known construction: Let X, Y be independent standard normals. Set Z = - (X+Y)/2? Check: Cov(X,Z) = -1/2, Var(Z) = 1/2, so Corr = -1/2 / sqrt(1*1/2) = -1/√2 ≈ -0.707. Not -1/2. Actually, the correct construction is: Let X, Y, Z be such that X+Y+Z=0 and Var(X)=Var(Y)=Var(Z). Then Corr = -1/2. For example, take X, Y i.i.d. with mean 0 and variance 1, and set Z = -X - Y. Then Var(Z)=2, so not equal variances. To have equal variances, we need a different approach. The standard example: Let X, Y, Z be three random variables with X+Y+Z=0 and Var(X)=Var(Y)=Var(Z)=1. Then Cov(X,Y) = -1/2, etc. One way: Let X and Y be independent? No, if X+Y+Z=0 and variances equal, then Cov(X,Y) = -1/2. So we can take any two variables with covariance -1/2 and set Z = -X-Y. But then Var(Z)=Var(X)+Var(Y)+2Cov(X,Y) = 1+1+2(-1/2)=1. So if we take X and Y with Var=1 and Cov=-1/2, then Z=-X-Y has Var=1 and Cov(X,Z)=Cov(X,-X-Y)=-Var(X)-Cov(X,Y)=-1+1/2=-1/2. Similarly for Y,Z. So we just need X and Y with correlation -1/2. For example, let X ~ N(0,1) and Y = -0.5 X + sqrt(3)/2 * W, where W ~ N(0,1) independent. Then Corr(X,Y) = -0.5. Then set Z = -X - Y. This works.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by clarifying that you need to transform uniform samples into normal samples, then present at least two methods: the Box-Muller transform and the inverse CDF method. Discuss the trade-offs between them, such as computational efficiency and numerical stability, and mention that the Ziggurat algorithm is a more advanced option used in practice.
Pro tip: Mention that while Box-Muller is simple, it can be slow due to trigonometric functions, and the inverse CDF method requires an accurate approximation of the probit function. For production systems, the Ziggurat algorithm is often preferred for its speed and accuracy.
Confirm that you need to generate standard normal samples using only a Uniform(0,1) generator, and that the method should be exact or approximate.
Explain that by generating two independent uniform variables U1 and U2, you can compute Z0 = sqrt(-2 ln U1) * cos(2π U2) and Z1 = sqrt(-2 ln U1) * sin(2π U2), which are independent standard normal.
Describe that if you can compute the inverse of the standard normal CDF (probit function), then X = Φ^{-1}(U) yields a standard normal sample. Mention that approximations like the Beasley-Springer-Moro algorithm are used.
Highlight that Box-Muller is easy to implement but uses trigonometric functions, while inverse CDF is efficient if a good approximation is available. Also mention the Ziggurat algorithm as a faster alternative.
Summarize that the choice depends on requirements: Box-Muller for simplicity, inverse CDF for speed if approximation is acceptable, and Ziggurat for high performance.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.