Problem Sets: Random Variables and Discrete Distributions (随机变量及其分布)

Which of the following function is indeed a probability density function?

• \( f(x) = \begin{cases} e^{-2x}, & x > 0 \
  0, & \text{otherwise} \end{cases} \)

• \( f(x) = \begin{cases} \sin x, & x \in [0, \pi] \
  0, & \text{otherwise} \end{cases} \)

• \( f(x) = \begin{cases} x^{2} - 1, & -1 < x < 1 \
  0, & \text{otherwise} \end{cases} \)

• \( f(x) = \begin{cases} \frac{1}{x^{2}}, & x > 1 \
  0, & \text{otherwise} \end{cases} \)

• \( f(x) = \begin{cases} \cos x, & x \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \
  0, & \text{otherwise} \end{cases} \)

• \( f(x) = \begin{cases} 1 - x, & -1 < x < 1 \
  0, & \text{otherwise} \end{cases} \)

• \( f(x) = \begin{cases} \frac{1}{x^{3}}, & x > 0 \
  0, & \text{otherwise} \end{cases} \)

• \( f(x) = \begin{cases} 2e^{-2x}, & x > 0 \
  0, & \text{otherwise} \end{cases} \)

• \( f(x) = \begin{cases} 1/x, & x > 0 \
  0, & \text{otherwise} \end{cases} \)

• \( f(x) = \begin{cases} e^{-2x}, & x > 0 \
  0, & \text{otherwise} \end{cases} \)

• \( f(x) = \begin{cases} x^{3} - 1, & -1 < x < 1 \
  0, & \text{otherwise} \end{cases} \)

Suppose that the random variable \(X\) has a continuous distribution with the following p.d.f.: \(f(𝐱) = \frac{1}{2}e^{-\lvert x\rvert}\), for \(-\infty < x <\infty\). Determine the value \(x_0\) such that \(F(𝐱_𝟎) = 0.9\), where \(F(x)\) is the c.d.f. of \(X\).

Assume that the cumulative distribution function of the random variable \( X \) is \( F(x), (-\infty < x < \infty) \), then \( \Pr(X > 5) = \)?

Three children are skating around the perimeter of a circle in an ice rink of radius 40 meters. Assume that their locations around the perimeter of the circle are independent, continuous random variables, each with constant density. Their mother comes to the door of the ice rink (located at a fixed position on perimeter of the circle). When she appears, what is the expected distance around the perimeter from her to the farthest of the three children?

• What is the probability that a random student passes the exam?

• What score is the 87.5 percentile of the distribution?

The lifetime (in years) of a music player, before it permanently fails, is a random variable with density \( f(x) = \begin{cases} \frac{1}{3}e^{-\frac{1}{3}x}, & x > 0, \
0, & \text{otherwise}. \end{cases} \) Consider 2 music players that are assumed to have independent lifetimes. What is the probability that the lifetime of one is longer than that of the other?

A person commutes from his home to the company for work every morning. Suppose that there are two routes can be chosen. The first route seems shorter but prone to traffic jam, leading to following p.d.f. of the corresponding commuting time (denoted by \( X \) mins): \( f(x) = \begin{cases} \frac{1}{5\sqrt{2\pi}}e^{-(x-40)^{2}/200}, & x > 40, \
0, & x \leq 40. \end{cases} \) The second route is longer, but the traffic is not congested, leading to the following p.d.f. of the corresponding commuting time (denoted by \( Y \) mins): \( f(y) = \begin{cases} \frac{1}{2\sqrt{2\pi}}e^{-(y-50)^{2}/72}, & y > 50, \
0, & y \leq 50. \end{cases} \) Hints: \( \Phi(2) \approx 0.9772, \Phi(2.5) \approx 0.9938, \Phi(1.5) \approx 0.9332, \Phi(1.25) \approx 0.8944 \).

• If that person leaves home 60 mins early, which route is more likely to result in being late for work?

• If that person leaves home 55 mins early, which route is more likely to result in being late for work?

Suppose that \(X_1\) and \(X_2\) are i.i.d. random variables, and that each has the uniform distribution on the interval \([0, 1]\). What is \(\Pr(X_1^{2} + X_2^{2} \leq 1)\)?

Consider the following joint p.d.f.’s for the random variables \( X \) and \( Y \). Select the one where \( X \) and \( Y \) are not independent.

• \( f(x,y) = 4x^{2}y^{3} \)

• \( f(x,y) = 1/2(x^{2}y + xy^{2}) \)

• \( f(x,y) = 6e^{-3x-2y} \)

• \( f(x,y) = e^{-x-2y} \)

Suppose that the cumulative distribution function (c.d.f.) of a random variable ( X ) is \( F(x) = \begin{cases} Ae^{-\frac{x}{2}} + B, & x \geq 0 \
\frac{1}{2}x + \frac{1}{2}, & -1 \leq x < 0 \
0, & x < -1 \end{cases} \). What is the value of \( A \) and \( B \)?

Suppose \( X \) and \( Y \) are lifetimes of two component in a computer. Suppose further that in the population of all computers, lifetime \( X \) and \( Y \) are distributed according to the following joint p.d.f. \( f(x,y) = \begin{cases} ke^{-(x+4y)}, & x > 0, y > 0, \
0, & \text{otherwise}. \end{cases} \)

• Determine the constant \( k \).

• Find the marginal p.d.f. of \( X \) and \( Y \).

• Find the joint c.d.f. of \( X \) and \( Y \).

• Compute the probability \( \Pr(X + Y > 4) \).

• Determine the p.d.f. of \( Z = X + Y \).

Assume \( X_1, X_2, X_3 \) are independent of each other and follow the Poisson distribution with \( \lambda = 3 \), let \( Y = \frac{1}{3}(X_1 + X_2 + X_3) \), then \( E(Y^{2}) = \)

Suppose that \( X \) and \( Y \) are random variables such that \( \text{Var}(X) = 2, \text{Var}(Y) = 5 \), \( X \) and \( Y \) are independent. What is the value of \( \text{Var}(2X - 3Y)\)?

Suppose that the random variables \( X, Y \) are independent, \( A = 2X + Y \), \( B = 2X - Y \), \( \text{Var}(X) = \text{Var}(Y) = 1 \). Compute \( \text{Cov}(A, B) \)?

Randomly mark two points \(X\) and \(Y\) on interval \([0,1]\). What is the expected length between these two points?

Suppose that \( X \) and \( Y \) are random variables such that \( \text{Var}(X) = 1, \text{Var}(Y) = 3 \), \( X \) and \( Y \) are independent. What is the value of \( \text{Var}(3X + 2Y)\)?

Suppose that \( X \sim B(1, 0.5) \) and \( Y \sim B(1, 0.5) \). If \( E(XY) = 0.5 \), what is the value of the correlation \( \rho(X,Y) \)?

Data was taken on height and weight from the entire population of 700 mountain gorillas living in the Democratic Republic of Congo (see the following table). Let \( X \) encode the weight, taking the values of a randomly chosen gorilla: 0, 1, 2 for light, average, and heavy, respectively. Likewise, let \( Y \) encode the height, taking values 0 and 1 for short and tall, respectively.

• Determine the joint p.m.f. of \( X \) and \( Y \) and the marginal p.m.f.’s of \( X \) and \( Y \).

• Are \( X \) and \( Y \) independent?

• Find the covariance of \( X \) and \( Y \).

• Find the correlation of \( X \) and \( Y \).

Suppose that a continuous random variable \( X \) has distribution with the following cumulative distribution function (c.d.f.): \( F(x) = \begin{cases} 1 - e^{-3x}, & x > 0 \
0, & \text{otherwise} \end{cases} \)

• Find the p.d.f. of \( X \);

• Find \( \Pr(X > 2) \);

• Find the \( E(X) \) and \( \text{Var}(X) \).

• Find the quantile function \( F^{-1}(p) \)

Suppose that the joint p.d.f. \( f(x,y) \) for the random variables \( X \) and \( Y \) is given as follows. \( f(x,y) = c \exp\left( -\frac{1}{2}(x^{2} + y^{2}) \right), x, y \in \mathbb{R} \).

• Determine the constant \( c \).

• Find the marginal p.d.f. of \( X \) and \( Y \), respectively; and determine whether \( X \) and \( Y \) are independent.

• Compute the \( E(3X + 5) \), \( \text{Var}(3Y + 5) \) and \( \text{Cov}(3X + 5, 3Y + 5) \).

• Let \( Z = X + Y \), find the p.d.f. of \( Z \).

Let both \( X \) and \( Y \) range \([0, 3]\) and joint density \( f_{X,Y}(x,y) = kxy \).

• Find the constant \( k \).

• Find the marginal probability density function for \( X \) and \( Y \), respectively.

• Find the marginal cumulative distribution function for \( X \) and \( Y \), respectively.

• Find the 30th percentile of \( X \) and \( Y \), respectively.

• Compute \( E(X) \), \( \text{Var}(X) \), \( E(Y) \), \( \text{Var}(Y)\)

• Compute \( \text{Cov}(X,Y) \) and correlation coefficient \( \rho(X,Y) \)

• Compute conditional probability density function \( f_{X \mid Y}(x \mid Y = 3) \)

Suppose that the joint p.d.f. of two random variables \( X \) and \( Y \) is as follows. \( f(x,y) = \begin{cases} ke^{-(2x+3y)}, & x > 0, y > 0 \
0, & \text{otherwise} \end{cases} \)

• Determine the constant \( k \).

• Find the marginal p.d.f. of \( X \) and \( Y \), respectively.

• Find the joint c.d.f. of \( X \) and \( Y \).

• Compute the probability \( \Pr(X+Y>6) \) and \( \Pr(Y>5 \mid X=3) \).

• Determine whether \( X \) and \( Y \) are independent.

• Compute \( E(2X + 6Y + 5) \) and \( \text{Var}(3Y + 7) \).

• Determine the p.d.f. of \( Z = X + Y \).

Suppose that the joint p.d.f. of two random variables \( X \) and \( Y \) is as follows \(f(x,y) = \begin{cases} cx^{2}, & 0 < x < 1, 0 \leq y \leq 1 - x^{2},
0, & \text{otherwise}. \end{cases} \). Determine

• The value of the constant \( c \).

• \( \Pr(Y \leq 1 - X) \)

• \( \Pr\left(Y \leq \frac{1}{2} \mid X = \frac{1}{2}\right) \)

Suppose that the random variable \( X \) is independent of the random variable \( Y \). Their probability density functions are \( f_X(x) = \begin{cases} 2e^{-2x}, & x > 0 \\ 0, & x \leq 0 \end{cases} \) and \( f_Y(y) = \begin{cases} \frac{1}{4}, & 0 < y < 4 \\ 0, & \text{otherwise} \end{cases} \), respectively. What is the value of \( \text{Var}(X + Y) \)?

Suppose that the discrete random variables \( X \) and \( Y \) have a joint discrete distribution with the following joint p.f. (p.m.f.) \( \Pr(X = i, Y = j) = m(i + j) \), where \( i = 1, 2 \) and \( j = 2, 3 \). What’s the value of \( m \)? ( C )

Suppose that the joint p.d.f. \( f(x,y) \) for the random variables \( X \) and \( Y \) is given as follows \( f(x,y) = \begin{cases} cy, & x^{2} < y < 1, \
0, & \text{otherwise}. \end{cases} \)

• Find the constant \( c \).

• Find \( \Pr(Y < X) \).

• Find the marginal p.d.f. of \( X \) and \( Y \) respectively, and determine whether \( X \) and \( Y \) are independent.

• Compute \( E(XY) \).

• Find the conditional p.d.f. of \( Y \) given that \( X = \frac{1}{2} \).

Suppose \( X \) is a random variable with c.d.f. \( F(x) = \begin{cases} 0, & \text{for } x < 0, \
x(2 - x), & \text{for } 0 \leq x \leq 1, \
1, & \text{for } x > 1. \end{cases} \)

• Find \( E(X) \)

• Find \( \Pr(X < 0.4) \)

Suppose that the joint p.d.f. of two random variables \( X \) and \( Y \) is as follows. \( f(x,y) = \begin{cases} 2e^{-(2x+y)}, & x > 0, y > 0 \
0, & \text{otherwise} \end{cases} \)

• Find the marginal p.d.f. of \( X \) and \( Y \), respectively.

• Find the marginal c.d.f. of \( X \) and \( Y \), respectively.

• Compute the probability \( \Pr(X+Y>5) \) and \( \Pr(Y>5\mid X=2) \).

• Determine whether \( X \) and \( Y \) are independent.

• Compute \( E(2X + 3Y + 5) \) and \( \text{Var}(2Y + 5) \)

• Determine the p.d.f. of \( Z = X + Y \)