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 $\text{Var}(X)$.

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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 $\text{Var}(X)$.

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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 $\text{Var}(X)$.

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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 $\text{Var}(X)$.

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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 $\text{Var}(X)$.

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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: $\text{Var}(X) = M_X''(0) - [M_X'(0)]^2 = 0.24$

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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: $\text{Var}(X) = M_X''(0) - [M_X'(0)]^2 = 0.49$

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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: $\text{Var}(X) = M_X''(0) - [M_X'(0)]^2 = 0.71$

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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: $\text{Var}(X) = M_X''(0) - [M_X'(0)]^2 = 0.49$

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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: $\text{Var}(X) = M_X''(0) - [M_X'(0)]^2 = 1.36$

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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.