1. (a) Two properties of a discrete probability distribution are: 1. Each probability $P(x)$ satisfies $0 \leq P(x) \leq 1$. 2. The sum of all probabilities equals 1, i.e., $\sum P(x) = 1$. (b) Difference between Bernoulli and Binomial distribution: - Bernoulli distribution models a single trial with two outcomes (success/failure). - Binomial distribution models the number of successes in $n$ independent Bernoulli trials. (c) Moment Generating Function (MGF) is useful because: - It uniquely characterizes the distribution. - It helps find moments (mean, variance) by differentiation. (d) To prove if $X \sim \text{Binomial}(m,p)$ and $Y \sim \text{Binomial}(n,p)$ are independent, then $X+Y \sim \text{Binomial}(m+n,p)$: 1. Given $M_X(s) = (pe^s + 1 - p)^m$ and $M_Y(s) = (pe^s + 1 - p)^n$. 2. Since $X$ and $Y$ are independent, $M_{X+Y}(s) = M_X(s) \times M_Y(s)$. 3. Substitute: $M_{X+Y}(s) = (pe^s + 1 - p)^m \times (pe^s + 1 - p)^n = (pe^s + 1 - p)^{m+n}$. 4. This is the MGF of a Binomial$(m+n,p)$ distribution. (e) For $X \sim \text{Exponential}(\lambda)$ with PDF $f_X(x) = \lambda e^{-\lambda x} d(x)$: 1. MGF is $M_X(s) = E[e^{sX}] = \int_0^\infty e^{sx} \lambda e^{-\lambda x} dx = \frac{\lambda}{\lambda - s}$ for $s < \lambda$. 2. Moments: $E[X^k] = \frac{k!}{\lambda^k}$. 2. (a) Types of kurtosis: - Mesokurtic: Normal distribution, kurtosis = 3. - Leptokurtic: Heavy tails, kurtosis > 3 (e.g., t-distribution). - Platykurtic: Light tails, kurtosis < 3 (e.g., uniform distribution). (b) Standard error of an estimator is the standard deviation of its sampling distribution, measuring estimator variability. (c) Complications in Maximum Likelihood Estimation (MLE): - May have multiple local maxima. - Requires large samples for accuracy. - Can be computationally intensive. (d) Bayesian estimation parameter involves updating prior beliefs with data to get posterior distribution. (e) Find quartiles and deciles for wages: 1. Calculate cumulative frequencies. 2. Locate positions for $Q_1 = 0.25 \times 65 = 16.25$, $Q_2 = 0.5 \times 65 = 32.5$, $Q_3 = 0.75 \times 65 = 48.75$. 3. Locate deciles $D_1 = 0.1 \times 65 = 6.5$, $D_3 = 0.3 \times 65 = 19.5$. 4. Use interpolation within class intervals to find exact values. Final answers: (a) Properties: $0 \leq P(x) \leq 1$, $\sum P(x) = 1$. (b) Bernoulli: single trial; Binomial: sum of trials. (c) MGF helps find moments. (d) $X+Y \sim \text{Binomial}(m+n,p)$ proven by MGF multiplication. (e) $M_X(s) = \frac{\lambda}{\lambda - s}$, $E[X^k] = \frac{k!}{\lambda^k}$. 2(a) Kurtosis types: mesokurtic, leptokurtic, platykurtic. (b) Standard error measures estimator variability. (c) MLE complications: local maxima, sample size, computation. (d) Bayesian estimation updates prior with data. (e) Quartiles and deciles found by cumulative frequency and interpolation.