Find accurate and reliable answers to your questions on IDNLearn.com. Discover in-depth answers from knowledgeable professionals, providing you with the information you need.

15. The probability distribution of a discrete random variable [tex]\( X \)[/tex] is given below:

[tex]\[
\begin{tabular}{|c|c|c|}
\hline
X & -1 & 0 \\
\hline
P(X) & 0.1 & 0.05 \\
\hline
\end{tabular}
\][/tex]

i. Find the moment generating function of [tex]\( X \)[/tex].

ii. Determine the mean and variance of [tex]\( X \)[/tex] using the mgf.


Sagot :

Alright, let’s solve the problem step-by-step.

### Step 1: Correct the Table of the Probability Distribution
First, let’s rewrite the properly formatted table for the given discrete random variable [tex]\( X \)[/tex].
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline X & -1 & 0 & 2 & 5 & 6 \\ \hline P(X) & 0.1 & 0.05 & 0.15 & 0.4 & 0.3 \\ \hline \end{array} \][/tex]

This table shows the values that [tex]\( X \)[/tex] can take and their respective probabilities.

### Step 2: Verify the Probability Distribution
The probabilities should sum to 1 to confirm that this is a valid probability distribution. Adding these probabilities:
[tex]\[ 0.1 + 0.05 + 0.15 + 0.4 + 0.3 = 1 \][/tex]
This confirms the table represents a valid probability distribution.

### Step 3: Calculate the Mean ([tex]\(\mu\)[/tex])
The mean (or expected value) is given by:
[tex]\[ \mu = E[X] = \sum (x \cdot P(x)) \][/tex]
Substituting the given values:
[tex]\[ \mu = (-1 \cdot 0.1) + (0 \cdot 0.05) + (2 \cdot 0.15) + (5 \cdot 0.4) + (6 \cdot 0.3) \][/tex]
[tex]\[ = -0.1 + 0 + 0.3 + 2 + 1.8 \][/tex]
[tex]\[ = 4.0 \][/tex]

So, the mean is [tex]\( \mu = 4.0 \)[/tex].

### Step 4: Calculate the Second Moment
The second moment about the origin is:
[tex]\[ E[X^2] = \sum (x^2 \cdot P(x)) \][/tex]
Substituting the given values:
[tex]\[ E[X^2] = ((-1)^2 \cdot 0.1) + (0^2 \cdot 0.05) + (2^2 \cdot 0.15) + (5^2 \cdot 0.4) + (6^2 \cdot 0.3) \][/tex]
[tex]\[ = (1 \cdot 0.1) + (0 \cdot 0.05) + (4 \cdot 0.15) + (25 \cdot 0.4) + (36 \cdot 0.3) \][/tex]
[tex]\[ = 0.1 + 0 + 0.6 + 10 + 10.8 \][/tex]
[tex]\[ = 21.5 \][/tex]

### Step 5: Calculate the Variance ([tex]\(\sigma^2\)[/tex])
The variance is given by:
[tex]\[ \sigma^2 = E[X^2] - (E[X])^2 \][/tex]
Substituting the values:
[tex]\[ \sigma^2 = 21.5 - (4.0)^2 \][/tex]
[tex]\[ = 21.5 - 16 \][/tex]
[tex]\[ = 5.5 \][/tex]

So, the variance is [tex]\( \sigma^2 = 5.5 \)[/tex].

### Step 6: Find the Moment Generating Function (MGF)
The moment generating function [tex]\( M_X(t) \)[/tex] for a discrete random variable [tex]\( X \)[/tex] is given by:
[tex]\[ M_X(t) = E[e^{tX}] = \sum e^{tx}P(x) \][/tex]

For this random variable [tex]\( X \)[/tex]:
[tex]\[ M_X(t) = (e^{-t} \cdot 0.1) + (e^{0 \cdot t} \cdot 0.05) + (e^{2t} \cdot 0.15) + (e^{5t} \cdot 0.4) + (e^{6t} \cdot 0.3) \][/tex]

### Step 7: Evaluate the MGF at [tex]\( t = 0 \)[/tex]
By definition, [tex]\( M_X(0) \)[/tex] should be 1, as:
[tex]\[ M_X(0) = \sum e^{0 \cdot x} P(x) = \sum 1 \cdot P(x) = \sum P(x) = 1 \][/tex]
So,
[tex]\[ M_X(0) = 1.0 \][/tex]

### Summary:
- Mean ([tex]\(\mu\)[/tex]): [tex]\( 4.0 \)[/tex]
- Variance ([tex]\(\sigma^2\)[/tex]): [tex]\( 5.5 \)[/tex]
- MGF evaluated at [tex]\( t=0 \)[/tex]: [tex]\( 1.0 \)[/tex]

Thus, we have successfully determined the mean, variance, and moment generating function [tex]\( M_X(t) \)[/tex] of the discrete random variable [tex]\( X \)[/tex].
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your questions deserve precise answers. Thank you for visiting IDNLearn.com, and see you again soon for more helpful information.