The probability P(n) that an event characterized by a probability p occurs n times in N trials is given by the binomial distribution Consider a case where p << 1 and N >> 1.

Several approximations can then be made to reduce Eq. (1) to a simpler form. Using the result that In (1 - p) -p, show that (1 - p)^N_n e^-Np. Show that N!/(N A- n)! N^n. Use Stirling's approximation (Appendix A of Baierlein). Hence show that Eq. (1) reduces to W(n) lambda^n/n! e^-lambda where lambda = Np is the mean number of events.

Horizontal axis with 1, 2, and 3 standard deviations above the mean are 51.6, 56.2, 60.8

Horizontal axis with 1, 2, and 3 standard deviations below the mean are 42.4, 37.8, 33.2

Step-by-step explanation:

A data set has a normal distribution with a mean of 47 and a standard deviation of 4.6

From this information, you have to scale the horizontal axis with the mean of this distribution and values at 1, 2, and 3 standard deviations above and below the mean.

Horizontal axis with 1, 2, and 3 standard deviations above the mean :

1 standard deviation above the mean ⇒ M + SD

⇒ 47 + 4.6 = 51.6

2 standard deviations above the mean ⇒ M + 2SD

⇒ 47 + (2 × 4.6) = 56.2

3 standard deviations above the mean ⇒ M + 3SD

⇒ 47 + (3 × 4.6) = 60.8

Horizontal axis with 1, 2, and 3 standard deviations below the mean :

1 standard deviation below the mean ⇒ M - SD

⇒ 47 - 4.6 = 42.4

2 standard deviations below the mean ⇒ M - 2SD

⇒ 47 - (2 × 4.6) = 37.8

3 standard deviations below the mean ⇒ M - 3SD

⇒ 47 - (3 × 4.6) = 33.2