Final answer:
To find probabilities and the standard deviation in an exponential distribution with a mean time of 2.6 minutes, we can use the exponential cumulative distribution function (CDF). The CDF formula is 1 - e^(-λx), where λ is the rate parameter. We can use the CDF to find the probabilities in parts (a), (b), and (c) of the question. The standard deviation can be calculated using the formula σ = 1/λ.
Step-by-step explanation:
To solve this problem, we can use the cumulative distribution function (CDF) of the exponential distribution. The CDF of an exponential distribution with rate parameter λ is given by 1 - e^(-λx), where x is the time. Since the mean time between hits is 2.6 minutes, we can calculate the rate parameter λ as 1/2.6. Now let's address each part of the question:
a. To find the probability that the next hit will occur within the next 4 minutes, we need to calculate P(x <= 4) using the exponential CDF. Plugging in the rate parameter λ = 1/2.6 and x = 4 into the CDF formula, we get P(x <= 4) = 1 - e^(-1/2.6 * 4).
b. To find the probability that the next hit will occur within the next 3 and 7 minutes, we need to calculate P(3 <= x <= 7). Again, we can use the exponential CDF with λ = 1/2.6 to find P(x <= 7) and P(x <= 3), then subtract them to find the desired probability.
c. To find the probability that the next hit will occur after the next 5 minutes, we need to calculate P(x > 5). We can use 1 - P(x <= 5) to find this probability.
d. To find the standard deviation of the distribution, we can use the formula σ = 1/λ, where λ is the rate parameter. Plugging in our calculated λ = 1/2.6, we can find the standard deviation.