Q1(a)

## [1] 0.5328072

Answer: 0.5328
Explanation: This is the probability that X lies between -5 and 10 for a normal distribution with mean 5 and variance 100.

Q1(b)

## [1] 0.4788891

Answer: 0.4789
Explanation: This is the probability that the absolute value of X exceeds 8, combining the lower and upper tails.

Q1(c)

## [1] 5

Answer: 5
Explanation: Since a normal distribution is symmetric, the value of c where P(X > c) = 0.5 equals the mean (5).

Q2

## [1] 0.9605087

Answer: 0.9605
Explanation: Using the Central Limit Theorem, this gives the probability that the sum of 100 die rolls is less than 380.

Q3

Answer:
\[ f(x_1, x_2, x_3) = \frac{10!}{x_1!x_2!x_3!}\left(\frac{1}{3}\right)^{10}, \quad x_1 + x_2 + x_3 = 10. \]
Explanation: This is a multinomial distribution with n = 10 and equal probabilities of 1/3 for each player.

Q4(a)

Answer:
\[ f(x, y) = 4e^{-x - 4y}, \quad x, y \ge 0. \]
Explanation: The joint density function is found by differentiating the given CDF with respect to both x and y.

Q4(b)

Answer:
\[ f_X(x) = e^{-x}, \quad f_Y(y) = 4e^{-4y}. \]
Explanation: The marginal densities are obtained by integrating the joint density over the other variable.

Q4(c)

Answer: X and Y are independent.
Explanation: Because \(f(x,y) = f_X(x)f_Y(y)\), independence is verified.

Q5(a)

## [1] 0.17 0.17 0.30 0.24 0.12
## [1] 0.10 0.29 0.38 0.17 0.06

Answer: Marginal of X₁ = 0.17, 0.17, 0.3, 0.24, 0.12
Marginal of X₂ = 0.1, 0.29, 0.38, 0.17, 0.06
Explanation: Marginals are obtained by summing across rows (for X₁) and columns (for X₂).

Q5(b)

## [1] 0.1 0.3 0.3 0.2 0.1

Answer: 0.1, 0.3, 0.3, 0.2, 0.1
Explanation: This gives the conditional distribution P(X₁ | X₂ = 0).

Q5(c)

## [1] 1.029563

Answer: 1.0296
Explanation: The standard deviation of X₂ is computed from its marginal probabilities.

Q6(a)

Answer:
\[ f_X(x) = x + \frac{1}{2}, \quad f_Y(y) = y + \frac{1}{2}. \]
Explanation: The marginals are found by integrating f(x, y) = x + y over y and x respectively.

Q6(b)

Answer:
\[ f_{X|Y}(x|y) = \frac{x + y}{y + 0.5}. \]
Explanation: The conditional density is f(x, y)/f_Y(y) for 0 ≤ x, y ≤ 1.

Q6(c)

## [1] -0.006944444

Answer: -0.0069
Explanation: The covariance measures how X and Y vary together; it’s positive, indicating a direct relationship.

Q6(d)

## [1] -0.09090909

Answer: -0.0909
Explanation: The correlation coefficient is the standardized covariance between X and Y.

Q7

Answer: Var(X + Y) is smaller than Var(X − Y).
Explanation: Negative correlation reduces the variance of a sum but increases the variance of a difference.

Q8

## [1] 209

Answer: 209
Explanation: Variance of (X − 4Y + 5) accounts for scaling and covariance between X and Y.

Q9(a)

Answer:
\[ P(Z_1=z_1, Z_2=z_2) = \begin{cases} p^2 & z_1=z_2=1, \\ p(1-p) & z_1 \ne z_2, \\ (1-p)^2 & z_1=z_2=0. \end{cases} \]
Explanation: This is the joint PMF of two independent Bernoulli variables.

Q9(b)

## $EY
## [1] 0.25
## 
## $VarY
## [1] 0.1875
## 
## $EY_Z1_0
## [1] 0

Answer: E(Y) = 0.25, Var(Y) = 0.19, E(Y | Z₁ = 0) = 0
Explanation: Y = Z₁Z₂, so Y = 1 only when both variables are 1; variance and conditional expectations follow Bernoulli logic.

Q9(c)

Answer: Y and Z₁ are not independent.
Explanation: Because Y = Z₁Z₂ depends directly on Z₁, the value of Z₁ influences Y.