Answer:
a) The probability that X is at most 2
= P(X ≤ 2) = 0.25
b) The expected value of X = 3.125
Step-by-step explanation:
The probability distribution is given as
f(x) = (2x - 1)/16, for x = 1,2,3,4
a) The probability that X is at most 2
= P(X ≤ 2) = P(X = 1) + P(X = 2)
P(X = 1) = f(x=1) = [(2×1) - 1]/16 = 0.0625
P(X = 2) = f(x=2) = [(2×2) - 1]/16 = 0.1875
P(X ≤ 2) = P(X = 1) + P(X = 2) = 0.0625 + 0.1875 = 0.25
b) The expected Value of X
Expected value is given as
E(X) = Σ xᵢpᵢ
xᵢ = each variable
pᵢ = probability of each variable
x = 1,2,3,4
But we need the corresponding probabilities for each variable
P(X = 1) = f(x=1) = 0.0625
P(X = 2) = f(x=2) = 0.1875
P(X = 3) = f(x=3) = [(2×3) - 1]/16 = 0.3125
P(X = 4) = f(x=4) = [(2×4) - 1]/16 = 0.4375
The probability mass function can then be given in tabular form as
X | p(x)
1 | 0.0625
2 | 0.1875
3 | 0.3125
4 | 0.4375
E(X) = Σ xᵢpᵢ = (1×0.0625) + (2×0.1875) + (3×0.3125) + (4×0.4375)
= 0.0625 + 0.375 + 0.9375 + 1.75
= 3.125
Hope this Helps!!!
