Consider the function fy (x,y)=c(x² +y² ), a joint probability density function over 02.5 and Y <1.5)Are X and Y independent? You must demonstrate this using the definition.What is the correlation between X and Y?е.f.g.What is the probability that Y<1.5 given X=0?

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A First Course in Probability (10th Edition)

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Author:Sheldon Ross

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Chapter1: Combinatorial Analysis

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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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A student's grade in a certain course is modeled by the joint PDF below; where X is the grade of the lecture component, and Y is the grade of the lab component. 6x + 10y, 0< x < 0.5,0 < y < 0.5 [0, fx,y (x, y) = otherwise Determine the probability that a student's grade in the lecture component is greater than his/her grade in the lab component.
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Consider a group of people who have received treatment for a disease such as cancer. Let t be the survival time, the number of years a person lives after receiving treatment. The density function giving the distribution of t is p(t) = Ce-Ct for some positive constant C, and the cumulative distribution function is P(t) = , p(x)dx. Think carefully about what the practical meaning of P(t) is, being sure that you can put it into words. a) The survival function, S(t), is the probability that a randomly selected person survives for at least t years. Find a formula for S(t). b) Suppose that a patient has a 80 percent chance of surviving at least 2 years. Find C. c) Using the value of C you found in (b), find the probability that the patient survives up to (that is, less than or equal to) 1 years. d) Using the value of C you found in (b), find the the mean survival time for patients with this survival function, in years.
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Consider the function fy (x,y)=c(x² +y² ), a joint probability density function over 0 2.5 and Y <1.5)