Question

Consider The Following Exponential Probability Density Function. F(X) = 1 4 E−X/4 For X ≥ 0 1) Write The Formula For P(X ≤ X0).

Answer 1

To find the probability P(X ≤ X_0) for an exponential probability density function f(x) = ⅔e^{-x/4}, integrate f(x) from 0 to X_0, yielding the CDF P(X ≤ X_0) = 1 - e^{-X_0/4}.

Step-by-step explanation:

The exponential probability density function given can be written as f(x) = ⅔e^{-x/4} for x ≥ 0, where the decay rate is ⅔ or 0.25. To find the probability that X is less than or equal to a certain value X0, we need to integrate the probability density function from 0 to X0, which yields the cumulative distribution function (CDF). The formula for the CDF is P(X ≤ X0) = 1 - e^{-x/4}.

To find the probability P(X ≤ X0), we would calculate the integral of f(x) from 0 to X0, giving:

P(X ≤ X0) = ∫0X0 (⅔e^{-x/4}) dx = 1 - e^{-X0/4}

This formula gives us the probability that the random variable X will take on a value less than or equal to X0 in an exponential distribution.

Answer 2

The formula for P(X ≤ X0) in exponential distribution is given by P(X ≤ X0) = 1 - e^(-0.25 * X0).

Step-by-step explanation:

Probability in Exponential Distribution

To find the formula for P(X ≤ X0), we need to use the cumulative distribution function (CDF) of the exponential distribution. The CDF for an exponential distribution with decay parameter m is given by P(X ≤ x) = 1 - e^(-mx). In this case, we have X ~ Exp(0.25), so the formula becomes P(X ≤ X0) = 1 - e^(-0.25 * X0).

Example:

If X0 is 2, then P(X ≤ 2) = 1 - e^(-0.25 * 2) = 1 - e^(-0.5) ≈ 0.3935.