Let X be a random variable with p.d.f 𝑓(𝑥) = {23 , 𝑤ℎ𝑒𝑛 𝑥 = 113 , 𝑤ℎ𝑒𝑛 𝑥 = 20, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒Find themoment generating function , mean and standard deviation

To find the moment generating function (MGF), mean, and standard deviation for the given probability distribution function (p.d.f.), we will follow these steps:

1. Break Down the Problem

1. Identify the values and probabilities associated with the random variable $X$.
2. Calculate the moment generating function $M_X(t)$.
3. Compute the mean $E[X]$.
4. Calculate the variance and standard deviation $\sigma$.

2. Relevant Concepts

The moment generating function $M_X(t)$ is defined as: $M_X(t) = E[e^{tX}] = \sum_{i} p_i e^{t x_i}$ Where $p_i$ is the probability of $X = x_i$.

The mean $E[X]$ can be calculated as: $E[X] = \sum_{i} p_i x_i$

The variance $\text{Var}(X)$ is computed using: $\text{Var}(X) = E[X^2] - (E[X])^2$ Where $E[X^2] = \sum_{i} p_i x_i^2$.

The standard deviation $\sigma$ is the square root of the variance: $\sigma = \sqrt{\text{Var}(X)}$

3. Analysis and Detail

1. Identifying Values and Probabilities: Let's denote:

   • $x_1 = 13$, $p_1 = \frac{2}{3}$
   • $x_2 = 20$, $p_2 = \frac{1}{3}$

2. Moment Generating Function Calculation: $M_X(t) = \left(\frac{2}{3}\right)e^{13t} + \left(\frac{1}{3}\right)e^{20t}$

3. Mean Calculation: $E[X] = \left(\frac{2}{3}\right)(13) + \left(\frac{1}{3}\right)(20) = \frac{26}{3} + \frac{20}{3} = \frac{46}{3}$

4. Second Moment Calculation: $E[X^2] = \left(\frac{2}{3}\right)(13^2) + \left(\frac{1}{3}\right)(20^2) = \left(\frac{2}{3}\right)(169) + \left(\frac{1}{3}\right)(400) = \frac{338}{3} + \frac{400}{3} = \frac{738}{3}$

5. Variance Calculation: $\text{Var}(X) = E[X^2] - (E[X])^2 = \frac{738}{3} - \left(\frac{46}{3}\right)^2 = \frac{738}{3} - \frac{2116}{9}$ Converting $\frac{738}{3}$ to have a common denominator with $\frac{2116}{9}$: $\frac{738}{3} = \frac{2214}{9}$ Therefore, $\text{Var}(X) = \frac{2214}{9} - \frac{2116}{9} = \frac{98}{9}$ And the standard deviation $\sigma$ is: $\sigma = \sqrt{\text{Var}(X)} = \sqrt{\frac{98}{9}} = \frac{\sqrt{98}}{3} = \frac{7\sqrt{2}}{3}$

4. Verify and Summarize

The moment generating function, mean, variance, and standard deviation have been correctly computed.

Final Answer

• Moment Generating Function: $M_X(t) = \frac{2}{3} e^{13t} + \frac{1}{3} e^{20t}$
• Mean: $E[X] = \frac{46}{3}$
• Standard Deviation: $\sigma = \frac{7\sqrt{2}}{3}$