Answer:
a) Expected Value for distribution A, E(X) = 3.020
Expected Value for distribution B, E(X) = 0.980
b) Standard deviation of distribution A = 1.157
Standard deviation of distribution B = 1.157
c) In distribution A, the bigger values of x have a higher probability of occurring than the values of distribution B (whose smaller values of x have a higher chance of occurring, hence, the expected value for distribution A is more than that of distribution B.
But according to the corresponding distribution of values, the two distributions have the same exact spread, A in ascending order (with higher values with bigger probability) and B in descending order (lower values have higher probabilities). But the same spread regardless, hence, the standard deviation which shows how data values spread around the mean (centre point) of a distribution is the same for the two distributions.
Step-by-step explanation:
Expected values is given as
E(X) = Σ xᵢpᵢ
where xᵢ = each possible sample space
pᵢ = P(X=xᵢ) = probability of each sample space occurring.
Distributions A and B is given by
X P(X) X P(x)
0 0.04 0 0.47
1 0.09 1 0.25
2 0.15 2 0.15
3 0.25 3 0.09
4 0.47 4 0.04
For distribution A
E(X) = Σ xᵢpᵢ = (0×0.04) + (1×0.09) + (2×0.15) + (3×0.25) + (4×0.47) = 3.02
For distribution B
E(X) = Σ xᵢpᵢ = (0×0.47) + (1×0.25) + (2×0.15) + (3×0.09) + (4×0.04) = 0.98
b) Standard deviation = √(variance)
But Variance is given by
Variance = Var(X) = Σx²p − μ²
where μ = E(X)
For distribution A
Σx²p = (0²×0.04) + (1²×0.09) + (2²×0.15) + (3²×0.25) + (4²×0.47) = 10.46
Variance = Var(X) = 10.46 - 3.02² = 1.3396
Standard deviation = √(1.3396) = 1.157
For distribution B
Σx²p = (0²×0.47) + (1²×0.25) + (2²×0.15) + (3²×0.09) + (4²×0.04) = 2.30
Variance = Var(X) = 2.30 - 0.98² = 1.3396
Standard deviation = √(1.3396) = 1.157
c) In distribution A, the bigger values of x have a higher probability of occurring than the values of distribution B (whose smaller values of x have a higher chance of occurring, hence, the expected value for distribution A is more than that of distribution B.
But according to the corresponding distribution of values, the two distributions have the same exact spread, A in ascending order (with higher values with bigger probability) and B in descending order (lower values have higher probabilities). But the same spread regardless, hence, the standard deviation which shows how data values spread around the mean (centre point) of a distribution is the same for the two distributions.
