Consider the probability function. (Give your answers exactly.)
P(x) =
5 − x
10
, for x = 1, 2, 3, 4
(a) Find the mean.
(b) Find the standard deviation.
What is the domain of the function?
P(x)=10 for x=1,2,3,4
and = 5-x otherwise?
What is the 10 doing there, please?
I don't know, is the answer 5-10=5 for mean????
Can you please type in the definition of the function exactly as you are given? As it is, it is unclear as to what the function is.
Is the 10 a denominator of the fraction
(5-x)/10?
Sorry, but this is the way it is written, that is why I do not understand it.
To find the mean of a probability function, we need to multiply each possible outcome by its corresponding probability and sum them up.
(a) The mean (μ) is calculated as follows:
μ = ∑(x * P(x))
Given the probability function P(x) = (5 - x)/10 for x = 1, 2, 3, 4, we can substitute these values into the formula to calculate the mean.
μ = (1 * P(1)) + (2 * P(2)) + (3 * P(3)) + (4 * P(4))
= (1 * (5 - 1)/10) + (2 * (5 - 2)/10) + (3 * (5 - 3)/10) + (4 * (5 - 4)/10)
= (1 * 4/10) + (2 * 3/10) + (3 * 2/10) + (4 * 1/10)
= (4/10) + (6/10) + (6/10) + (4/10)
Summing up the fractions, we get:
= 20/10
= 2
Therefore, the mean of the probability function is 2.
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To find the standard deviation, we need to calculate the variance first. The variance (σ^2) is calculated as the average of squared deviations from the mean.
(b) The variance is calculated as follows:
σ^2 = ∑[(x - μ)^2 * P(x)]
Given the probability function P(x) = (5 - x)/10 for x = 1, 2, 3, 4, and the mean μ = 2, we can substitute these values into the formula to calculate the variance.
σ^2 = [(1 - 2)^2 * P(1)] + [(2 - 2)^2 * P(2)] + [(3 - 2)^2 * P(3)] + [(4 - 2)^2 * P(4)]
= [(-1)^2 * (5 - 1)/10] + [(0)^2 * (5 - 2)/10] + [(1)^2 * (5 - 3)/10] + [(2)^2 * (5 - 4)/10]
= (1 * 4/10) + (0 * 3/10) + (1 * 2/10) + (4 * 1/10)
Summing up the fractions, we get:
= 8/10
= 0.8
Therefore, the variance of the probability function is 0.8.
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Finally, the standard deviation (σ) is the square root of the variance.
σ = √σ^2
= √0.8
= 0.8944 (rounded to four decimal places)
Therefore, the standard deviation of the probability function is approximately 0.8944.