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.

To find the mean of a probability function, you need to calculate the weighted average of the possible outcomes, where the weights are given by the probabilities of each outcome.

(a) To find the mean, you can use the formula:

μ = ∑ (x * P(x))

where μ is the mean, x represents the possible outcomes, and P(x) represents the probability of each outcome.

For the given probability function:
P(x) = (5 - x) / 10

You can find the mean by plugging in the values of x and P(x) into the formula:

μ = (1 * P(1)) + (2 * P(2)) + (3 * P(3)) + (4 * P(4))

Substituting the values for P(x):

μ = (1 * (5 - 1)/10) + (2 * (5 - 2)/10) + (3 * (5 - 3)/10) + (4 * (5 - 4)/10)

μ = (1 * 4/10) + (2 * 3/10) + (3 * 2/10) + (4 * 1/10)

μ = 4/10 + 6/10 + 6/10 + 4/10

μ = 20/10

μ = 2

So, the mean of the given probability function is 2.

(b) To find the standard deviation, you can use the formula:

σ = √∑[(x - μ)^2 * P(x)]

where σ is the standard deviation, x represents the possible outcomes, μ is the mean, and P(x) represents the probability of each outcome.

Using the given probability function, P(x) = (5 - x) / 10, and the mean μ = 2, you can calculate the standard deviation as follows:

σ = √[(1 - 2)^2 * P(1) + (2 - 2)^2 * P(2) + (3 - 2)^2 * P(3) + (4 - 2)^2 * P(4)]

Substituting the values for P(x):

σ = √[(1 - 2)^2 * (5 - 1)/10 + (2 - 2)^2 * (5 - 2)/10 + (3 - 2)^2 * (5 - 3)/10 + (4 - 2)^2 * (5 - 4)/10]

σ = √[(1 - 2)^2 * 4/10 + (2 - 2)^2 * 3/10 + (3 - 2)^2 * 2/10 + (4 - 2)^2 * 1/10]

σ = √[(1^2 * 4/10) + (0^2 * 3/10) + (1^2 * 2/10) + (2^2 * 1/10)]

σ = √[4/10 + 0 + 2/10 + 8/10]

σ = √(14/10)

σ = √(7/5)

So, the standard deviation of the given probability function is √(7/5).