Quiz 2

The following work should be completed individually. Use of notes or textbooks is not allowed. You may use a scientific calculator, not a graphing calculator or phone app.

Show all work unless instructed otherwise.

Write either True or False for each of the following statements. No justification is required.

Let n, k be integers with 0 \leq k \leq n. Then {n\choose k} = {n \choose n - k}.

True.

Suppose X is a continuous random variable with pdf f(x) and cdf F(x). Then \displaystyle{\int_a^b f(x)\ dx = F(b) - F(a)}.

True.

Suppose X is a random variable taking values between 0 and 6. Then \E(X) = 3.

False.

Consider the joint distribution for X and Y below.

Joint Distribution X = 0 X = 1 X = 2 Y = 0 0.1 0.05 0.1 Y = 1 0.3 0.15 0.3

Find the marginal distributions for X and Y.

\Pr(X = 0) \amp = 0.1 + 0.3 = 0.4 \amp \Pr(Y = 0) \amp = 0.1 + 0.05 + 0.1 = 0.25 \Pr(X = 1) \amp = 0.05 + 0.15 = 0.2 \amp \Pr(Y = 1) \amp = 0.3 + 0.15 + 0.3 = 0.75 \Pr(X = 2) \amp = 0.1 + 0.3 = 0.4

Are X and Y independent?

Checking each cell in the table: \Pr(X = 0, Y = 0) \amp = 0.1 = (0.4)(0.25) = \Pr(X = 0)\Pr(Y = 0) \Pr(X = 1, Y = 0) \amp = 0.05 = (0.2)(0.25) = \Pr(X = 1)\Pr(Y = 0) \Pr(X = 2, Y = 0) \amp = 0.1 = (0.4)(0.25) = \Pr(X = 2)\Pr(Y = 0) \Pr(X = 0, Y = 1) \amp = 0.3 = (0.4)(0.75) = \Pr(X = 0)\Pr(Y = 1) \Pr(X = 1, Y = 1) \amp = 0.15 = (0.2)(0.75) = \Pr(X = 1)\Pr(Y = 1) \Pr(X = 2, Y = 1) \amp = 0.3 = (0.4)(0.75) = \Pr(X = 2)\Pr(Y = 1) So X, Y are independent.

Suppose a continuous random variable X taking values in [0, 1] has cdf F(x) = 2x^2 - x^4.

Find the pdf f(x).

f(x) = F'(x) = 4x - 4x^3.

Find \E(X).

\E(X) \amp = \int_0^1 x f(x)\ dx \amp = \int_0^1 x (4x - 4x^3)\ dx \amp = \int_0^1 4x^2 - 4x^4\ dx \amp = \frac{4x^2}{3} - \frac{4x^5}{5}\bigg|_0^1 \amp = \left(\frac{4}{3} - \frac{4}{5}\right) - (0) \amp = \frac{8}{15} \approx 0.533.