An engineer is studying early morning traffic patterns at a particular intersection. The observation period begins at 5:30 a.m. Let X denote the time of arrival of the first vehicle from the north-south direction; let Y denote the first arrival time from the east-west direction.Time is measured in fractions of an hour after 5:30 a.m. Assume that the density for (X, Y) is given by f(x, y) = 1/x 0 < y < x < 1 Verify that this is a joint density for a two-dimensional random variable. Find P(X < 0.5 and Y < 0.25). Find the marginal densities for X and Y. Find P(X < 0.5). Find P(Y < 0.25). Are X and Y independent? Explain. Find E(X), E(Y), E(XY) and Cov(X, Y).
An engineer is studying early morning traffic patterns at a particular intersection. The observation period begins at 5:30 a.m. Let X denote the time of arrival of the first vehicle from the north-south direction; let Y denote the first arrival time from the east-west direction.Time is measured in fractions of an hour after 5:30 a.m. Assume that the density for (X, Y) is given by f(x, y) = 1/x 0 < y < x < 1 Verify that this is a joint density for a two-dimensional random variable. Find P(X < 0.5 and Y < 0.25). Find the marginal densities for X and Y. Find P(X < 0.5). Find P(Y < 0.25). Are X and Y independent? Explain. Find E(X), E(Y), E(XY) and Cov(X, Y).
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 92E
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Question
An engineer is studying early morning traffic patterns at a particular intersection. The observation period begins at 5:30 a.m. Let X denote the time of arrival of the first vehicle from the north-south direction; let Y denote the first arrival time from the east-west direction.
Time is measured in fractions of an hour after 5:30 a.m. Assume that the density for (X, Y) is given by
f(x, y) = 1/x
0 < y < x < 1
- Verify that this is a joint density for a two-dimensional random variable.
- Find P(X < 0.5 and Y < 0.25).
- Find the marginal densities for X and Y.
- Find P(X < 0.5).
- Find P(Y < 0.25).
- Are X and Y independent? Explain.
- Find E(X), E(Y), E(XY) and Cov(X, Y).
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