Find moment generating functions (mgf) of the following probability distributions. Using mgf found in above; find mean and variance
The Correct Answer and Explanation is:
1. Bernoulli Distribution (X ∼ Bernoulli(p))
MGF:MX(t)=E[etX]=(1−p)⋅e0+p⋅et=1−p+petM_X(t) = E[e^{tX}] = (1 – p) \cdot e^{0} + p \cdot e^{t} = 1 – p + p e^{t}MX(t)=E[etX]=(1−p)⋅e0+p⋅et=1−p+pet
Mean and Variance:
- Mean: MX′(0)=pM_X'(0) = pMX′(0)=p
- Variance: MX′′(0)−(MX′(0))2=p(1−p)M_X”(0) – (M_X'(0))^2 = p(1 – p)MX′′(0)−(MX′(0))2=p(1−p)
2. Binomial Distribution (X ∼ Binomial(n, p))
MGF:MX(t)=(1−p+pet)nM_X(t) = \left(1 – p + p e^{t}\right)^nMX(t)=(1−p+pet)n
Mean and Variance:
- Mean: MX′(0)=npM_X'(0) = npMX′(0)=np
- Variance: np(1−p)np(1 – p)np(1−p)
3. Poisson Distribution (X ∼ Poisson(λ))
MGF:MX(t)=eλ(et−1)M_X(t) = e^{\lambda(e^{t} – 1)}MX(t)=eλ(et−1)
Mean and Variance:
- Mean: λ\lambdaλ
- Variance: λ\lambdaλ
4. Exponential Distribution (X ∼ Exponential(λ))
MGF:MX(t)=λλ−t,for t<λM_X(t) = \frac{\lambda}{\lambda – t}, \quad \text{for } t < \lambdaMX(t)=λ−tλ,for t<λ
Mean and Variance:
- Mean: 1λ\frac{1}{\lambda}λ1
- Variance: 1λ2\frac{1}{\lambda^2}λ21
5. Normal Distribution (X ∼ N(μ, σ²))
MGF:MX(t)=eμt+12σ2t2M_X(t) = e^{\mu t + \frac{1}{2} \sigma^2 t^2}MX(t)=eμt+21σ2t2
Mean and Variance:
- Mean: μ\muμ
- Variance: σ2\sigma^2σ2
Explanation
The moment generating function, abbreviated as mgf, is a useful tool in probability theory that helps characterize a distribution. It is defined as the expected value of the exponential function of a random variable, written mathematically as MX(t)=E[etX]M_X(t) = E[e^{tX}]MX(t)=E[etX]. The primary purpose of the mgf is to generate moments of a distribution, such as the mean and variance. By differentiating the mgf and evaluating it at zero, one can compute these important statistical properties.
For instance, the first derivative of the mgf at zero gives the mean of the distribution. The second derivative gives the second moment, and combining both allows us to find the variance. For discrete distributions like the Bernoulli and Binomial, the mgf simplifies the process of summing over possible outcomes. For continuous distributions like the Exponential and Normal, the mgf involves integrating over a probability density function.
The mgf is especially powerful because if two random variables have the same mgf, they have the same distribution, provided the mgf exists in an open interval around zero. This uniqueness property makes mgfs valuable in proving convergence in distribution, such as in the Central Limit Theorem.
Another key benefit of mgfs is their use in analyzing sums of independent random variables. If variables are independent, the mgf of their sum is the product of their individual mgfs. This simplifies computation and is used heavily in modeling and simulations.
Overall, the mgf is not just a theoretical construct but a practical tool for finding expected values, variances, and for establishing properties of distributions.
