SOLUTION: On the midnight shift, the number of patients with head trauma in an emergency room has the probability distribution shown below. x 0 1 2 3 4 5 Total P(x) .03 .42 .30 .19 .0

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Question 1196800: On the midnight shift, the number of patients with head trauma in an emergency room has the probability distribution shown below.

x 0 1 2 3 4 5 Total
P(x) .03 .42 .30 .19 .05 .01 1.00

(a) Calculate the mean and standard deviation. (Round your mean value to 2 decimal places and standard deviation to 3 decimal places.)

Answer by math_tutor2020(3817) (Show Source):
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Answers:
Mean = 1.84
Standard Deviation = 0.997

The mean is exact, while the standard deviation is approximate.

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Work Shown:

We'll use this table to compute the mean

x 0 1 2 3 4 5 Total
P(x) 0.03 0.42 0.3 0.19 0.05 0.01 1
x*P(x) 0 0.42 0.6 0.57 0.2 0.05 1.84

The x*P(x) row refers to multiplying each column of x and P(x) values.
Then we add up all of the x*P(x) values to get 1.84 which is the mean.

The Greek letter mu is often used as the mean
So we have mu = 1.84 and this value is useful to find the standard deviation.

x 0 1 2 3 4 5 Total
P(x) 0.03 0.42 0.3 0.19 0.05 0.01 1
X-mu -1.84 -0.84 0.16 1.16 2.16 3.16
(x-mu)^2 3.3856 0.7056 0.0256 1.3456 4.6656 9.9856
(x-mu)^2*P(x) 0.101568 0.296352 0.00768 0.255664 0.23328 0.099856 0.9944

The value 0.9944 in the bottom right corner represents the sum of the (x-mu)^2*P(x) values
This value also represents the variance of the probability distribution table.

Take the square root of the variance to get the standard deviation
standard deviation = sqrt(variance)
standard deviation = sqrt(0.9944)
standard deviation = 0.99719606898543
standard deviation = 0.997