To find the missing probability, we can use the fact that the sum of all probabilities in a probability distribution is equal to 1.
Given the probabilities provided, we can calculate the missing probability for x = 4 by subtracting the sum of the other probabilities from 1.
Sum of probabilities:
0.12 + 0.18 + 0.30 + 0.15 + 0.10 + 0.05 = 0.90
Missing probability:
1 – 0.90 = 0.10
So, the missing probability for x = 4 is 0.10.
To find the expected value, we multiply each value of x by its corresponding probability, and then sum up the results. Let’s calculate it:
Expected value (E) = (0 * 0.12) + (1 * 0.18) + (2 * 0.30) + (3 * 0.15) + (4 * 0.10) + (5 * 0.10) + (6 * 0.05)
E = 0 + 0.18 + 0.60 + 0.45 + 0.40 + 0.50 + 0.30 = 2.43
Therefore, the expected value from the table is 2.43.
To find the standard deviation, we need to calculate the variance first. The variance is the average of the squared differences between each x value and the expected value, weighted by their corresponding probabilities (Hayes, 2023).
Variance (Var) = [(0 – E)^2 * 0.12] + [(1 – E)^2 * 0.18] + [(2 – E)^2 * 0.30] + [(3 – E)^2 * 0.15] + [(4 – E)^2 * 0.10] + [(5 – E)^2 * 0.10] + [(6 – E)^2 * 0.05]
Var = [(0 – 2.43)^2 * 0.12] + [(1 – 2.43)^2 * 0.18] + [(2 – 2.43)^2 * 0.30] + [(3 – 2.43)^2 * 0.15] + [(4 – 2.43)^2 * 0.10] + [(5 – 2.43)^2 * 0.10] + [(6 – 2.43)^2 * 0.05]
Var = 0.7086 + 0.5922 + 0.2808 + 0.1047 + 0.0627 + 0.0627 + 0.0413 = 1.852
The standard deviation (SD) is the square root of the variance:
SD = √Var = √1.852 = 1.36
Therefore, the standard deviation from the table is approximately 1.36.
References
Hayes, A. (2023, March 14). What Is Variance in Statistics? Definition, Formula, and Example. Retrieved from https://www.investopedia.com/terms/v/variance.asp