1. **Problem statement:** Given independent random variables $X_1 \sim N(60,25)$, $X_2 \sim \text{Gam}(3.5,\sqrt{5})$, $X_3 \sim \text{Gam}(6,\sqrt{3})$, and $X_4 \sim N(50,16)$, identify the distributions of the following: - $Y_1 \sim N(70,34)$ - $\chi^2_{12}$ - $T_7$ - $F_{7,12}$ - $\text{Beta}(3.5,6)$ 2. **Step 1: Understanding the distributions** - Normal distribution $N(\mu, \sigma^2)$ has mean $\mu$ and variance $\sigma^2$. - Gamma distribution $\text{Gam}(\alpha, \beta)$ with shape $\alpha$ and scale $\beta$. - Chi-square distribution $\chi^2_k$ is a special case of Gamma with shape $k/2$ and scale 2. - Student's t-distribution $T_k$ arises from a normal variable divided by the square root of a scaled chi-square variable. - F-distribution $F_{d_1,d_2}$ is the ratio of two scaled independent chi-square variables. - Beta distribution $\text{Beta}(\alpha, \beta)$ is defined on [0,1] with shape parameters $\alpha$ and $\beta$. 3. **Step 2: Identify $Y_1 \sim N(70,34)$** - This is a normal distribution with mean 70 and variance 34. - It can be formed by linear combinations of independent normal variables. 4. **Step 3: Identify $\chi^2_{12}$** - $\chi^2_{12}$ is a chi-square distribution with 12 degrees of freedom. - It can be represented as the sum of squares of 12 independent standard normal variables. 5. **Step 4: Identify $T_7$** - $T_7$ is a Student's t-distribution with 7 degrees of freedom. - It can be constructed as $T_7 = \frac{Z}{\sqrt{W/7}}$ where $Z \sim N(0,1)$ and $W \sim \chi^2_7$ independent. 6. **Step 5: Identify $F_{7,12}$** - $F_{7,12}$ is an F-distribution with numerator degrees of freedom 7 and denominator degrees of freedom 12. - It can be represented as $F_{7,12} = \frac{(U/7)}{(V/12)}$ where $U \sim \chi^2_7$ and $V \sim \chi^2_{12}$ independent. 7. **Step 6: Identify $\text{Beta}(3.5,6)$** - Beta distribution with shape parameters 3.5 and 6. - Can be constructed from Gamma variables: if $X \sim \text{Gam}(3.5,1)$ and $Y \sim \text{Gam}(6,1)$ independent, then $\frac{X}{X+Y} \sim \text{Beta}(3.5,6)$. **Final answers:** - $Y_1 \sim N(70,34)$ is a normal distribution with mean 70 and variance 34. - $\chi^2_{12}$ is a chi-square distribution with 12 degrees of freedom. - $T_7$ is a Student's t-distribution with 7 degrees of freedom. - $F_{7,12}$ is an F-distribution with numerator 7 and denominator 12 degrees of freedom. - $\text{Beta}(3.5,6)$ is a Beta distribution with shape parameters 3.5 and 6. These distributions are standard and can be related to the given variables through known transformations and definitions.