The random variables X and Y have a joint PDF of the form fX,Y(x,y)=c⋅exp{−12(4x2−8x+y2−6y+13)}.
To find the constant (c) in the given joint probability density function (PDF), we need to integrate the PDF over its entire domain and set it equal to 1 since the PDF must integrate to 1 over the entire domain to be a valid probability distribution.
The given joint PDF is:
fX,Y(x, y) = c * exp(-12(4x^2 - 8x + y^2 - 6y + 13))
To find the constant (c), we integrate the PDF over the entire domain, which is the range of X and Y:
∫∫ fX,Y(x, y) dx dy
Integrating with respect to x first:
∫∫ c * exp(-12(4x^2 - 8x + y^2 - 6y + 13)) dx dy
To integrate this expression, we treat y as a constant and integrate with respect to x. The integral of exp(ax^2 +bx + c) with respect to x is given by:
√(π/a) * exp((b^2 - 4ac)/4a)
In our case, a = -48, b = -96x - 6y, and c = -12(4x^2 + y^2 - 6y + 13). Plugging these values into the formula:
∫ c * exp(-12(4x^2 - 8x + y^2 - 6y + 13)) dx
= c * √(π/(-48)) * exp(((-96x - 6y)^2 - 4*(-12(4x^2 + y^2 - 6y + 13))*-48)/(-192))
= c * (1/4) * √3π * exp((16x^2 + 48x + 4y^2 - 24y + 52)/3)
Now, we need to integrate this expression with respect to y:
∫∫ c * (1/4) * √3π * exp((16x^2 + 48x + 4y^2 - 24y + 52)/3) dx dy
To integrate this expression, we treat x as a constant and integrate with respect to y. The integral of exp(ay^2 + by + c) with respect to y is given by:
(√π/a) * exp((b^2 - 4ac)/(4a))
In our case, a = 4, b = -24y, and c = (16x^2 + 48x + 52)/3. Plugging these values into the formula:
∫ c * (1/4) * √3π * exp((16x^2 + 48x + 4y^2 - 24y + 52)/3) dx
= c * (1/4) * √3π * (√π/4) * exp((-36y^2 - 48y + (16x^2 + 48x + 52)/3)/(4*4))
Simplifying further:
= c * (1/16) * 3π * (√π/4) * exp((4x^2 + 12x - 3y^2 - 4y + 13)/3)
Now, we need to integrate this expression over the entire range of y. Since there is no specific information about the range of y, we assume it is from negative infinity to positive infinity:
∫∫ c * (1/16) * 3π * (√π/4) * exp((4x^2 + 12x - 3y^2 - 4y + 13)/3) dx dy
= c * (1/16) * 3π * (√π/4) * √(3/4) * (√π/3) * exp(13/3)
= c * (√π/64) * π * (√π/12) * exp(13/3)
= c * (√π^4/768) * π * exp(13/3)
= c * (√π^5/768) * exp(13/3)
To find c, we set the value of the integral equal to 1:
c * (√π^5/768) * exp(13/3) = 1
Solving for c:
c = 768 / (√π^5 * exp(13/3))
Therefore, the constant (c) in the joint PDF is 768 / (√π^5 * exp(13/3)).