Question 1

Suppose $$X$$ and $$Y$$ are independent continuous random variables with densities $$f_X(x)$$ and $$f_Y(y)$$, respectively. Which of the following correctly expresses $$P\{X \u003c Y\}$$?

Accepted Answer

$$P\{X \u003c Y\} = \int_{-\infty}^{\infty} F_X(y) f_Y(y) dy$$

Question 2

Let $$X$$ and $$Y be $$jointly continuous random variables with joint density $$f(x, y). $$The marginal density of $$X is $$given by:

Accepted Answer

$$f_X(x) = \int_{-\infty}^{\infty} f(x, y) dy$$

Question 3

Suppose $$X$$ and $$Y$$ are independent normal random variables with means $$\mu_X, \mu_Y$$ and variances $$\sigma_X^2$, \sigma_Y^2$. What is the distribution of X + Y$?

Accepted Answer

Normal with mean $$\mu_X + \mu_Y$$ and variance $$\sigma_X^2$ + \sigma_Y^2$

Question 4

Let $$X be a $$binomial random variable with parameters $$n$$ and $$p. $$Which of the following is the variance of $$X$$?

Accepted Answer

$$np(1-p)$$

Question 5

Suppose $$X$$ and $$Y$$ are jointly continuous with joint density $$f(x, y). $$The conditional density of $$X$$ given $$Y = y is$$:

Accepted Answer

$$f_{X|Y}(x|y) = \frac{f(x, y)}{f_Y(y)}$$

Question 6

A store receives a random number $$N of $$customers in a day, with $$E[N] = 50. $$Each customer spends an independent amount $$X_i$$ with $$E[X_i] = 8. $$What is the expected total amount spent in the store that day?

Accepted Answer

$$E[N] \cdot E[X_i] = 400$$

Question 7

Let $$X_1$$, \ldots, X_n$$ be independent and identically distributed random variables with mean $$\mu$$ and variance $$\$$sigma^2. $$The sample mean is $$\overline{X} = \frac{1}{n} \sum_{i=1}^n X_i. $$What is $$Var(\overline{X})$$?

Accepted Answer

$$\frac{\$$sigma^2$$}{n}$$

Question 8

Suppose $$X is a $$Poisson random variable with mean $$\lambda. $$What is the moment generating function (MGF) of $$X$$?

Accepted Answer

$$M(t) = \exp\{\lambda (e$$^{t}$$ - 1)\}$$

Question 9

Let $$X$$ and $$Y be $$random variables with $$Cov(X, Y) = 0. $$Which of the following must be true?

Accepted Answer

X and Y are uncorrelated

Question 10

Suppose $$X is a $$nonnegative integer-valued random variable. Which identity is correct for its expectation?

Accepted Answer

$$E[X] = \sum_{i=1}^{\infty} P\{X \geq i\}$$

Question 11

Let $$X$$ and $$Y be $$independent Poisson random variables with means $$\$$lambda_1$$ and $$\$$lambda_2. $$What is the distribution of $$X + Y$$?

Accepted Answer

Poisson with mean $$\$$lambda_1$$ + \$$lambda_2$$

Question 12

Suppose $$X$$ and $$Y$$ are jointly continuous random variables. Which of the following is always true?

Accepted Answer

$$E[X + Y] = E[X] + E[Y]$$

Question 13

Let $$X be a $$geometric random variable with parameter $$p. $$What is $$Var(X)$$?

Accepted Answer

$$\frac{1-p}{p^2$}$$

Question 14

Suppose $$X$$ and $$Y$$ are independent and uniformly distributed on $$(0, 1). $$What is the joint density of $$U = X + Y$$ and $$V = X/Y$$?

Accepted Answer

It can be found by the change of variables formula using the Jacobian determinant

Question 15

Let $$X_1$$, \ldots, X_n$$ be independent uniform (0$$, 1)$$ random variables. What is the expected value of the sample mean $$\overline{X}$$?

Accepted Answer

$$0.5$$

Question 16

Suppose $$X$$ and $$Y$$ are jointly continuous with joint density $$f(x, y). $$Which of the following is the correct formula for the expectation of a function $$g(X, Y)$$?

Accepted Answer

$$E[g(X, Y)] = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(x, y) f(x, y) dx dy$$

Jointly Distributed Random Variables, Expectation, and Properties: Study Notes

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