With an equation
p(X=x)=e to the-lambda *lambda to the x/x how do you isolate for lambda if you know p ?
Math assistance needed.
Take the ln of both sides.
To isolate lambda in the equation p(X=x) = e^(-lambda) * lambda^x / x, you need to use logarithms. Here's how you can do it step by step:
Step 1: Start with the equation p(X=x) = e^(-lambda) * lambda^x / x.
Step 2: Multiply both sides of the equation by x to remove the denominator:
x * p(X=x) = e^(-lambda) * lambda^x
Step 3: Take the natural logarithm (ln) of both sides of the equation to eliminate the exponential term:
ln(x * p(X=x)) = ln(e^(-lambda) * lambda^x)
Step 4: Use the logarithmic property ln(a * b) = ln(a) + ln(b) to split the right side of the equation:
ln(x) + ln(p(X=x)) = ln(e^(-lambda)) + ln(lambda^x)
Step 5: Simplify the equation using the logarithmic property ln(e^a) = a:
ln(x) + ln(p(X=x)) = -lambda + x * ln(lambda)
Step 6: Rearrange the equation to isolate lambda:
lambda = (ln(x) + ln(p(X=x))) / (-1 - ln(lambda))
Note: In most cases, it is not possible to solve this equation explicitly for lambda. Instead, you would need to use numerical methods or approximation techniques to estimate the value of lambda based on the known values of p and x.