Geologists estimate the time since the most recent cooling of a mineral by counting the number of uranium fission tracks on the surface of the mineral. A certain mineral specimen is of such an age that there should be an average of 6 tracks per cm2 of surface area. Assume that the number of tracks in an area follows a Poisson distribution. Let X represent the number of tracks counted in 1 cm2 of surface area. Compute P(X = 7). Round your answer to four decimal places. Compute P(X ≥ 3). Round your answer to four decimal places. Compute P(2 < X < 7). Round your answer to four decimal places. Compute σX, the standard deviation of X. Round your answer to four decimal places.

(a) The value of P (X = 7) is 0.1377.

(b) The value of P (X ≥ 3) is 0.9380.

(c) The value of P (2 < X < 7) is 0.5443.

(d) The standard deviation of X is 2.4495

Step-by-step explanation:

Let X = the number of tracks counted in 1 cm² of surface area.

It is provided that the average number of tracks is 6 per cm².

The random variable .

The probability function of a Poisson distribution is:

(a)

Compute the value of P (X = 7) as follows:

Thus, the value of P (X = 7) is 0.1377.

(b)

Compute the value of P (X ≥ 3) as follows:

P (X ≥ 3) = 1 - P (X < 3)

              = 1 - P (X = 0) - P (X = 1) - P (X = 2)

Thus, the value of P (X ≥ 3) is 0.9380.

(c)

Compute the value of P (2 < X < 7) as follows:

P (2 < X < 7) = P (X = 3) + P (X = 4) + P (X = 5) + P (X = 6)

Thus, the value of P (2 < X < 7) is 0.5443.

(d)

The standard deviation of a Poisson distribution is:

Compute the value of standard deviation of X as follows:

Thus, the standard deviation of X is 2.4495