The random variable follows the standard normal distribution . Given , the random variable follows the normal distribution . (1) Determine the distribution of ; (2) Compute .

Probability Theory Solution – Problem 60: Determine the distribution of ;

Question

The random variable $X$ follows the standard normal distribution $N(0,\ 1)$ . Given $X=x$ , the random variable $Y$ follows the normal distribution $N(x,\ 1)$ . (1) Determine the distribution of $Y$ ; (2) Compute $P(X Y\geqslant0)$ .

Step-by-step solution

1. Structural decomposition of the random variable From the given condition $Y|X=x \sim N(x, 1)$ , given $X=x$ , $Y$ equals $x$ plus a standard normal random variable. We can represent $Y$ structurally as: $Y = X + Z$ where $X \sim N(0, 1)$ , $Z \sim N(0, 1)$ , and $Z$ is independent of $X$ . (Justification: when $X$ is fixed at $x$ , $X+Z$ follows $N(x, 1)$ , matching the problem statement.)

2. Computing the expectation and variance of $Y$ Since $X$ and $Z$ are independent and both follow normal distributions, their linear combination $Y$ also follows a normal distribution. Expectation: $E[Y] = E[X + Z] = E[X] + E[Z] = 0 + 0 = 0$ Variance: $Var(Y) = Var(X + Z) = Var(X) + Var(Z)$ (by independence) $Var(Y) = 1 + 1 = 2$

3. Conclusion The random variable $Y$ follows a normal distribution with mean 0 and variance 2, i.e., $Y \sim N(0, 2)$ .

1. Reformulating the problem Substituting $Y = X + Z$ into the inequality $XY \geqslant 0$ : $P(XY \geqslant 0) = P(X(X + Z) \geqslant 0) = P(X^2 + XZ \geqslant 0)$

2. Using the geometric approach (polar coordinates) Since $X \sim N(0, 1)$ , $Z \sim N(0, 1)$ and they are independent, the joint density of $(X, Z)$ has rotational symmetry. Introduce polar coordinates: $X = R \cos \theta$ , $Z = R \sin \theta$ , where $R > 0$ , $\theta \in [0, 2\pi)$ , and the density is uniform over $\theta$ .

3. Analyzing the angular range satisfying the condition Substituting into $X^2 + XZ \geqslant 0$ : $R^2 \cos^2 \theta + R^2 \cos \theta \sin \theta \geqslant 0$ Dividing by $R^2 > 0$ : $\cos \theta (\cos \theta + \sin \theta) \geqslant 0$

Boundary points: $\cos \theta = 0$ ( $\theta = \frac{\pi}{2}, \frac{3\pi}{2}$ ) and $\cos \theta + \sin \theta = 0$ ( $\tan \theta = -1$ , $\theta = \frac{3\pi}{4}, \frac{7\pi}{4}$ ).

Sign analysis on $[0, 2\pi)$ : * $(0, \frac{\pi}{2})$ : product $> 0$ . Arc length: $\frac{\pi}{2}$ . * $(\frac{\pi}{2}, \frac{3\pi}{4})$ : product $< 0$ . * $(\frac{3\pi}{4}, \frac{3\pi}{2})$ : product $> 0$ . Arc length: $\frac{3\pi}{4}$ . * $(\frac{3\pi}{2}, \frac{7\pi}{4})$ : product $< 0$ . * $(\frac{7\pi}{4}, 2\pi)$ : product $> 0$ . Arc length: $\frac{\pi}{4}$ .

4. Computing the total probability Total arc length: $L = \frac{\pi}{2} + \frac{3\pi}{4} + \frac{\pi}{4} = \frac{3\pi}{2}$ $P(XY \geqslant 0) = \frac{L}{2\pi} = \frac{3}{4}$

Final answer

(1) $Y \sim N(0, 2)$ (2) $P(XY \geqslant 0) = \frac{3}{4}$

Marking scheme

The following is the complete marking scheme for this probability theory problem.

I. Checkpoints (max 7 pts total)

Part 1: Distribution of $Y$ (3 points) [additive]

1 pt: Model construction. Write the structural decomposition $Y = X + Z$ and state that $Z$ is independent of $X$ (or $Z \sim N(0,1)$ ), or write the correct total probability formula / marginal density integral $f_Y(y) = \int f_{Y|X}(y|x)f_X(x)dx$ , or write the characteristic function product form.
1 pt: Derivation. Use independence to derive $E[Y]=0$ and $Var(Y)=2$ (must show the $1+1=2$ computation), or correctly complete the integral / completing-the-square computation, or simplify the characteristic function.
1 pt: Final conclusion. Explicitly state $Y \sim N(0, 2)$ (with parameters specified).
*Note: If the result $N(0,2)$ is stated directly without any derivation, this part receives 0 points.*

Part 2: Computing $P(XY \ge 0)$ (4 points)

Score exactly one chain | take the maximum subtotal among chains; do not add points across chains.

Chain A: Structural decomposition and polar coordinates (standard approach)
1 pt: Inequality transformation. Transform $XY \ge 0$ into a form involving independent variables, such as $X(X+Z) \ge 0$ or $X^2 + XZ \ge 0$ . [additive]
2 pts: Region analysis. Correctly analyze the region in the $(X, Z)$ plane satisfying the condition (e.g., using polar coordinates to obtain total arc length $3\pi/2$ , or correctly discussing the range of $Z$ for $X>0$ and $X<0$ ). [additive]
1 pt: Result. Obtain probability $3/4$ (or $0.75$ ). [additive]
Chain B: Bivariate normal geometric method (Sheppard's theorem / correlation coefficient)
1 pt: Distribution identification. Explicitly state that $(X, Y)$ follows a bivariate normal distribution. [additive]
1 pt: Correlation computation. Correctly compute the correlation coefficient $\rho = \frac{1}{\sqrt{2}}$ (or covariance $Cov(X,Y)=1$ with correct variances). [additive]
1 pt: Geometric formula application. Apply the arcsine formula $P = \frac{1}{2} + \frac{\arcsin \rho}{\pi}$ , or derive using the geometric angle properties of the density contour ellipses. [additive]
1 pt: Result. Obtain probability $3/4$ . [additive]
Chain C: Joint density integration method
1 pt: Integral setup. Use symmetry to write $2P(X>0, Y>0)$ or write a double integral with the correct joint density $f(x,y)$ . [additive]
2 pts: Integral evaluation. Correctly evaluate the definite integral via substitution (e.g., $u=x, v=y/x$ ) or polar coordinates. [additive]
1 pt: Result. Obtain probability $3/4$ . [additive]

Total (max 7)

II. Zero-credit items

In Part 1, merely listing formulas (e.g., the normal density formula) without substituting the problem conditions ( $Y|X$ ) or performing calculations.
In Part 1, stating $Y \sim N(0,2)$ directly without any justification.
In Part 2, incorrectly assuming $X$ and $Y$ are independent, obtaining the result $1/2$ .
In Part 2, incorrectly assuming $X$ and $Y$ are perfectly positively correlated ( $Y=kX, k>0$ ), obtaining the result $1$ .

III. Deductions

Missing logical justification (-1): In Part 1, using $Var(X+Z) = Var(X) + Var(Z)$ without mentioning "independence" or "zero covariance" anywhere.
Imprecise inequality handling (-1): In Part 2, dividing both sides of $X(X+Z) \ge 0$ by $X$ without considering the reversal of the inequality sign when $X<0$ (even if the correct region is obtained by symmetry).
Arithmetic error (-1): Completely correct approach but errors in simple arithmetic or trigonometric values (e.g., $\arctan(-1)$ ).
*Note: Total score after deductions cannot be less than 0; if multiple solution methods are present, only the highest-scoring one is graded.*

Probability Theory – Problem 60: Determine the distribution of ;