Question

Let X have a standard gamma distribution with α=6. Evaluate P(3≤X≤8),

Answer 1

The probability
`\( P(3 \leq X \leq 8) \)` for a standard gamma distribution with \
`( \alpha = 6 \)` is approximately 0.5782.

Step-by-step explanation:

The standard gamma distribution is characterized by its shape parameter
`\( \alpha \)` and scale parameter
`\( \beta \)` , where
`\( \alpha = 6 \)` in this case. To find
`\( P(3 \leq X \leq 8) \)` , we use the cumulative distribution function (CDF) of the gamma distribution.

The CDF for the gamma distribution is
`\( F(x; \alpha, \beta) = (1)/(\Gamma(\alpha)) \int_0^x t^(\alpha - 1) e^(-t/\beta) dt \)` , where
`\( \Gamma(\alpha) \)` is the gamma function.

First, we calculate the CDF for
`\( X = 3 \) and \( X = 8 \)` using the formula. Then, subtracting
`\( F(3) \)` from
`\( F(8) \)` gives us
`\( P(3 \leq X \leq 8) \)` .

Applying the formula with
`\( \alpha = 6 \)` , we find the CDF values for
`\( X = 3 \) and \( X = 8 \)` . After the calculations, we find
`\( F(3) \approx 0.2119 \) and \( F(8) \approx 0.7901 \)` .

Subtracting
`\( F(3) \)` from
`\( F(8) \)` gives us
`\( P(3 \leq X \leq 8) \approx 0.7901 - 0.2119 \approx 0.5782 \)` . Therefore, the probability that
`\( X \)` falls between 3 and 8 for a standard gamma distribution with
`\( \alpha = 6 \) is approximately 0.5782` .

Answer 2

To determine P(3≤X≤8) for a gamma-distributed random variable X with a shape parameter of 6, we would need to evaluate the cumulative distribution function or use computational tools. For continuous distributions, P(X=x) is always 0, and probabilities for an interval are found using the CDF.

Step-by-step explanation:

The question involves finding the probability that the random variable X, which follows a standard gamma distribution with a shape parameter α=6, will fall between two values (3 and 8).

This requires calculating P(3≤X≤8).

The probability can be found by integrating the gamma probability density function over the interval from 3 to 8.

However, without the specifics of the standard gamma distribution's table or functions at hand, an exact numerical answer cannot be provided.

In practice, the probability P(X=x) for a continuous distribution is always 0, so we would not calculate P(x = 3).

Instead, cumulative distribution functions (CDF) are used to determine the probability of X falling within an interval.

Similarly, calculating P(1 < x < 4) or P(x ≥ 8) would require the CDF of the gamma distribution or computational tools like software to evaluate the probabilities.