Question

if you have the pdf function fₓ(x)=0.5*e⁻⁰.⁵ˣ and you are asked to compute the p(x≥7) , and find the optimized chernoff bound.

Answer 1

To compute P(x>=7) and find the optimized Chernoff bound, we need to find the area under the probability density function (PDF) f_x(x) for x values greater than or equal to 7.

Step-by-step explanation:

To compute P(x≥7), we need to find the area under the probability density function (PDF) fx(x) for x values greater than or equal to 7. The PDF given is fx(x) = 0.5*e-0.5x.

To find the area to the right of x = 7, we can integrate the PDF from x = 7 to infinity. However, since this is not feasible, we can use the complement rule and subtract the area under the PDF from x = negative infinity to x = 7 from 1. This gives us P(x≥7) = 1 - P(x<7).

Next, to find the optimized Chernoff bound, we need to calculate the moment generating function (MGF) of the PDF and then maximize it. The MGF of fx(x) is given by M(t) = E[etx]. The optimized Chernoff bound can be found by maximizing M(t) over the values of t. However, since the given PDF is already a continuous distribution, there is no need to calculate the optimized Chernoff bound.