Give me 5 questions with Specific PMFS  (e.g. x: 0, 1 P(x) = .... ) wher i must find E(X) or Var(X) after derriving MGF, put the oslutoons below

Here are five questions involving specific probability mass functions (PMFs) where you need to find the expected value \( E(X) \) and variance \( \text{Var}(X) \) after deriving the moment generating function (MGF):

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### 1. **Question 1**:
Given the following PMF:
- \( x: 0, 1 \)
- \( P(X=0) = 0.6 \)
- \( P(X=1) = 0.4 \)

Find:
- (a) The MGF \( M_X(t) \).
- (b) \( E(X) \) and \( \text{Var}(X) \).

---

### 2. **Question 2**:
Given the following PMF:
- \( x: 0, 1, 2 \)
- \( P(X=0) = 0.3 \)
- \( P(X=1) = 0.5 \)
- \( P(X=2) = 0.2 \)

Find:
- (a) The MGF \( M_X(t) \).
- (b) \( E(X) \) and \( \text{Var}(X) \).

---

### 3. **Question 3**:
Given the following PMF:
- \( x: -1, 0, 1 \)
- \( P(X=-1) = 0.2 \)
- \( P(X=0) = 0.5 \)
- \( P(X=1) = 0.3 \)

Find:
- (a) The MGF \( M_X(t) \).
- (b) \( E(X) \) and \( \text{Var}(X) \).

---

### 4. **Question 4**:
Given the following PMF:
- \( x: 1, 2, 3 \)
- \( P(X=1) = 0.2 \)
- \( P(X=2) = 0.5 \)
- \( P(X=3) = 0.3 \)

Find:
- (a) The MGF \( M_X(t) \).
- (b) \( E(X) \) and \( \text{Var}(X) \).

---

### 5. **Question 5**:
Given the following PMF:
- \( x: 0, 2, 4 \)
- \( P(X=0) = 0.3 \)
- \( P(X=2) = 0.4 \)
- \( P(X=4) = 0.3 \)

Find:
- (a) The MGF \( M_X(t) \).
- (b) \( E(X) \) and \( \text{Var}(X) \).

---

## Solutions

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### **Solution 1**:
1. **MGF**:
   \[
   M_X(t) = 0.6e^{0t} + 0.4e^{1t} = 0.6 + 0.4e^t
   \]
2. **Expected value**:
   \[
   E(X) = M_X'(0) = 0.4
   \]
3. **Variance**:
   \[
   \text{Var}(X) = M_X''(0) - [M_X'(0)]^2 = 0.24
   \]

---

### **Solution 2**:
1. **MGF**:
   \[
   M_X(t) = 0.3e^{0t} + 0.5e^{1t} + 0.2e^{2t} = 0.3 + 0.5e^t + 0.2e^{2t}
   \]
2. **Expected value**:
   \[
   E(X) = M_X'(0) = 0.9
   \]
3. **Variance**:
   \[
   \text{Var}(X) = M_X''(0) - [M_X'(0)]^2 = 0.49
   \]

---

### **Solution 3**:
1. **MGF**:
   \[
   M_X(t) = 0.2e^{-t} + 0.5e^{0t} + 0.3e^{t} = 0.2e^{-t} + 0.5 + 0.3e^{t}
   \]
2. **Expected value**:
   \[
   E(X) = M_X'(0) = 0.1
   \]
3. **Variance**:
   \[
   \text{Var}(X) = M_X''(0) - [M_X'(0)]^2 = 0.71
   \]

---

### **Solution 4**:
1. **MGF**:
   \[
   M_X(t) = 0.2e^{1t} + 0.5e^{2t} + 0.3e^{3t} = 0.2e^t + 0.5e^{2t} + 0.3e^{3t}
   \]
2. **Expected value**:
   \[
   E(X) = M_X'(0) = 2.1
   \]
3. **Variance**:
   \[
   \text{Var}(X) = M_X''(0) - [M_X'(0)]^2 = 0.49
   \]

---

### **Solution 5**:
1. **MGF**:
   \[
   M_X(t) = 0.3e^{0t} + 0.4e^{2t} + 0.3e^{4t} = 0.3 + 0.4e^{2t} + 0.3e^{4t}
   \]
2. **Expected value**:
   \[
   E(X) = M_X'(0) = 2.2
   \]
3. **Variance**:
   \[
   \text{Var}(X) = M_X''(0) - [M_X'(0)]^2 = 1.36
   \]

---

### Follow-up Questions:
1. Can you derive the MGF for a binomial distribution with \( n = 2 \) and \( p = 0.5 \)?
2. What is the relation between the MGF and the characteristic function of a random variable?
3. How does the second derivative of the MGF at \( t = 0 \) relate to variance?
4. How would you compute the MGF for a Poisson distribution?
5. Can you find \( E(X) \) and \( \text{Var}(X) \) for a geometric distribution using the MGF?

**Tip**: When deriving the MGF, remember that it is the expectation of \( e^{tX} \), so carefully substitute the values of \( X \) and compute the sum.

Give me 5 questions with Specific PMFS (e.g. x: 0, 1 P(x) = .... ) where I must find E(X) or Var(X) after deriving MGF, put the solutions below.

Explore five problems involving specific probability mass functions (PMFs) where you derive the moment generating function (MGF) to find the expected value (E(X)) and variance (Var(X)). Detailed solutions provided, ideal for students in Grades 11-12 or undergraduate level.

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