"You are required to submit a substantial response to all questions and provide all necessary theorems and techniques for the solutions"Given the table below, answer the following questions:xP(x)00.1210.1820.3030.15450.1060.05 Complete the missing probability.What is the expected value from the table?Find the standard deviation

1. Complete the missing probability:

The sum of all probabilities in a probability distribution should be equal to 1. So, to find the missing probability, we add up all the given probabilities and subtract from 1.

0.12 + 0.18 + 0.30 + 0.15 + 0.10 + 0.05 = 0.90

So, the missing probability is 1 - 0.90 = 0.10

2. Expected value:

The expected value (E) of a random variable is the sum of the product of each outcome and its probability. So, we multiply each x value by its corresponding probability and sum them up.

E = (00.12) + (10.18) + (20.30) + (30.15) + (40.10) + (50.05) + (6*0.10) = 0 + 0.18 + 0.60 + 0.45 + 0.40 + 0.30 + 0.60 = 2.53

3. Standard deviation:

The standard deviation is a measure of the amount of variation or dispersion of a set of values. To find the standard deviation, we first need to find the variance. The variance is the expected value of the squared deviation of a random variable from its mean value.

First, calculate the squared deviation from the mean for each outcome, multiply by the corresponding probability, and sum up to find the variance.

Variance = Σ [ P(x) * (x - E)^2 ]

= (0.12*(0-2.53)^2) + (0.18*(1-2.53)^2) + (0.30*(2-2.53)^2) + (0.15*(3-2.53)^2) + (0.10*(4-2.53)^2) + (0.05*(5-2.53)^2) + (0.10*(6-2.53)^2)

Calculate the above expression to get the variance.

Finally, the standard deviation is the square root of the variance.