Question

A random variable mathrm{X} follows a distribution with probability density function:

[ f_{X}(x)=left\{begin{array}{cc} 4 e^{-k x} & 0

Answer 1

To find P(4 < x < 5), we can use the exponential distribution's cumulative distribution function (CDF). Calculate P(x < 5) and P(x < 4) using the CDF, then subtract P(x < 4) from P(x < 5) to get the desired probability.

Step-by-step explanation:

The random variable X follows an exponential distribution with a probability density function given by fX(x) = 4e-kx for 0 < x < ∞. To find P(4 < x < 5), we can use the cumulative distribution function (CDF) which gives the area to the left. Let's calculate it step by step:

1. First, let's find P(x < 5) using the CDF: P(x < x) = 1 - e-kx
2. Substituting x = 5 into the equation, we get P(x < 5) = 1 - e-5k
3. Next, let's find P(x < 4) using the CDF: P(x < x) = 1 - e-kx
4. Substituting x = 4 into the equation, we get P(x < 4) = 1 - e-4k

Therefore, P(4 < x < 5) = P(x < 5) - P(x < 4) = 1 - e-5k - (1 - e-4k) = e-4k - e-5k.