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4.5-1. Let X and Y have a bivariate normal distribution with parameters µx = -3, µy = 10, o? = p = 3/5. Compute 25, o = 9, and %3D (a) P(-5 < X < 5). (b) P(-5 < X < 5|Y = 13). (c) P(7< Y < 16). %3D

4.5-1. Let X and Y have a bivariate normal distribution with parameters µx = -3, µy = 10, o? = p = 3/5. Compute 25, o = 9, and %3D (a) P(-5 < X < 5). (b) P(-5 < X < 5|Y = 13). (c) P(7< Y < 16). %3D

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4.5-1

**Exercises:** **4.5-1.** Let \( X \) and \( Y \) have a bivariate normal distribution with parameters \( \mu_X = -3 \), \( \mu_Y = 10 \), \(\sigma_X^2 = 25\), \(\sigma_Y^2 = 9\), and \(\rho = \frac{3}{5}\). Compute: - (a) \( P(-5 < X < 5) \). - (b) \( P(-5 < X < 5 \mid Y = 13) \). - (c) \( P(7 < Y < 16) \). - (d) \( P(7 < Y < 16 \mid X = 2) \). **4.5-2.** Show that the expression in the exponent of Equation 4.5-2 is equal to the function \( q(x, y) \) given in the text. **4.5-3.** Let \( X \) and \( Y \) have a bivariate normal distribution with parameters \( \mu_X = 2.8 \), \( \mu_Y = 110 \), \(\sigma_X^2 = 0.16\), \(\sigma_Y^2 = 100\), and \(\rho = 0.6\). Compute: - (a) \( P(108 < Y < 126) \). - (b) \( P(108 < Y < 126 \mid X = 3.2) \). **4.5-4.** Let \( X \) and \( Y \) have a bivariate normal distribution with \( \mu_X = 70 \), \(\sigma_X^2 = 100\), \( \mu_Y = 80 \), \(\sigma_Y^2 = 169\), and \(\rho = \frac{5}{13}\). Find: - (a) \( E(Y \mid X = 76) \). - (b) \( \text{Var}(Y \mid X = 76) \). - (c) \( P(Y \leq 86 \mid X = 76) \).
Transcribed Image Text:**Exercises:** **4.5-1.** Let \( X \) and \( Y \) have a bivariate normal distribution with parameters \( \mu_X = -3 \), \( \mu_Y = 10 \), \(\sigma_X^2 = 25\), \(\sigma_Y^2 = 9\), and \(\rho = \frac{3}{5}\). Compute: - (a) \( P(-5 < X < 5) \). - (b) \( P(-5 < X < 5 \mid Y = 13) \). - (c) \( P(7 < Y < 16) \). - (d) \( P(7 < Y < 16 \mid X = 2) \). **4.5-2.** Show that the expression in the exponent of Equation 4.5-2 is equal to the function \( q(x, y) \) given in the text. **4.5-3.** Let \( X \) and \( Y \) have a bivariate normal distribution with parameters \( \mu_X = 2.8 \), \( \mu_Y = 110 \), \(\sigma_X^2 = 0.16\), \(\sigma_Y^2 = 100\), and \(\rho = 0.6\). Compute: - (a) \( P(108 < Y < 126) \). - (b) \( P(108 < Y < 126 \mid X = 3.2) \). **4.5-4.** Let \( X \) and \( Y \) have a bivariate normal distribution with \( \mu_X = 70 \), \(\sigma_X^2 = 100\), \( \mu_Y = 80 \), \(\sigma_Y^2 = 169\), and \(\rho = \frac{5}{13}\). Find: - (a) \( E(Y \mid X = 76) \). - (b) \( \text{Var}(Y \mid X = 76) \). - (c) \( P(Y \leq 86 \mid X = 76) \).
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