Consider the signal x(t),which consists of single rectangular pulse of unif height, is symmetric about the origin, and has a total width T1.( a) sketch x(t).?

To sketch the signal x(t), we need to consider the given properties:

1. The signal is a single rectangular pulse.
2. The pulse is symmetric about the origin.
3. The pulse has a total width T1.

Let's start by drawing the x-axis and the y-axis.

Next, we draw a rectangular pulse centered at the origin. Since the pulse is symmetric, we can draw half of the pulse on one side of the origin and mirror it on the other side.

The width of the pulse is T1, so we draw the pulse from -T1/2 to T1/2 on the x-axis.

The height of the pulse is uniform, so we draw the pulse with a constant value on the y-axis.

The resulting sketch of x(t) would look like a rectangular pulse centered at the origin, with a total width of T1 and a uniform height.

To find the probability for the given scenarios, we can use the Poisson probability formula:

P(x;λ) = (e^(-λ) * λ^x) / x!

Where
λ = 12 (average number of incidents per year)
x = the number of occurrences for each scenario

a) Probability that exactly three will experience anaphylaxis:
P(3;12) = (e^(-12) * 12^3) / 3!
P(3;12) ≈ 0.1114 or 11.14%

b) Probability that at least two will experience anaphylaxis:
P(at least 2;12) = 1 - P(0;12) - P(1;12)
P(at least 2;12) = 1 - (e^(-12) * 12^0) / 0! - (e^(-12) * 12^1) / 1!
P(at least 2;12) ≈ 0.8647 or 86.47%

c) Probability that at most two will experience anaphylaxis:
P(at most 2;12) = P(0;12) + P(1;12) + P(2;12)
P(at most 2;12) = (e^(-12) * 12^0) / 0! + (e^(-12) * 12^1) / 1! + (e^(-12) * 12^2) / 2!
P(at most 2;12) ≈ 0.2642 or 26.42%

In a certain population an average of 13 new cases of  esophageal cancer are diagnosed each year. If the annual  incidence of esophageal cancer follows a Poisson  distribution, find the probability that in a given year the number of newly diagnosed cases of esophageal cancer  will be:A. Exactly 10 cases B. At least three cases C. No more than 3D. Between nine and 12, inclusiveE. Fewer than two

In a certain population an average of 13 new cases of  esophageal cancer are diagnosed each year. If the annual  incidence of esophageal cancer follows a Poisson  distribution, find the probability that in a given year the number of newly diagnosed cases of esophageal cancer  will be:A. Exactly 10 cases B. At least three cases C. No more than 3D. Between nine and 12, inclusiveE. Fewer than two