Find all moments mun = E[Xⁿ] of a random variable X whose moment generating function is S(t). Compute mu1, mu2, mu3, mu7

The n-th moment μₙ = E[Xⁿ] is given by:
μₙ = (dⁿ/dtⁿ) S(t) |_(t=0)
This means you take the n-th derivative of S(t) with respect to t, and then substitute t=0 into the resulting expression.

Since the specific form of the moment generating function S(t) is not provided, we cannot compute the numerical values for μ₁, μ₂, μ₃, and μ₇. However, we can express them in terms of the derivatives of S(t).

• For the first moment (mean): μ₁ = S'(0) (This is the first derivative of S(t) evaluated at t=0.)

• For the second moment: μ₂ = S''(0) (This is the second derivative of S(t) evaluated at t=0.)

• For the third moment: μ₃ = S'''(0) (This is the third derivative of S(t) evaluated at t=0.)

• For the seventh moment: μ₇ = S⁽⁷⁾(0) (This is the seventh derivative of S(t) evaluated at t=0.)

To get numerical values, you would need to provide the explicit function S(t).