1991 Paper 3 Q16

Year: 1991
Paper: 3
Question Number: 16

Course: UFM Statistics
Section: Bivariate data

Difficulty: 1700.0 Banger: 1504.3

probability bivariate data joint distribution marginal probability expectation summation discrete random variables integer constraints

Problem

The random variables \(X\) and \(Y\) take integer values \(x\) and \(y\) respectively which are restricted by \(x\geqslant1,\) \(y\geqslant1\) and \(2x+y\leqslant2a\) where \(a\) is an integer greater than 1. The joint probability is given by \[ \mathrm{P}(X=x,Y=y)=c(2x+y), \] where \(c\) is a positive constant, within this region and zero elsewhere. Obtain, in terms of \(x,c\) and \(a,\) the marginal probability \(\mathrm{P}(X=x)\) and show that \[ c=\frac{6}{a(a-1)(8a+5)}. \] Show that when \(y\) is an even number the marginal probability \(\mathrm{P}(Y=y)\) is \[ \frac{3(2a-y)(2a+2+y)}{2a(a-1)(8a+5)} \] and find the corresponding expression when \(y\) is off. Evaluate \(\mathrm{E}(Y)\) in terms of \(a\).

No solution available for this problem.

Rating Information

Difficulty Rating: 1700.0

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Banger Rating: 1504.3

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Problem source

The random variables $X$ and $Y$ take integer values $x$ and $y$
respectively which are restricted by $x\geqslant1,$ $y\geqslant1$
and $2x+y\leqslant2a$ where $a$ is an integer greater than 1. The
joint probability is given by 
\[
\mathrm{P}(X=x,Y=y)=c(2x+y),
\]
where $c$ is a positive constant, within this region and zero elsewhere.
Obtain, in terms of $x,c$ and $a,$ the marginal probability $\mathrm{P}(X=x)$
and show that 
\[
c=\frac{6}{a(a-1)(8a+5)}.
\]
Show that when $y$ is an even number the marginal probability $\mathrm{P}(Y=y)$
is 
\[
\frac{3(2a-y)(2a+2+y)}{2a(a-1)(8a+5)}
\]
and find the corresponding expression when $y$ is off. 

Evaluate $\mathrm{E}(Y)$ in terms of $a$.

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