Week 10 Worksheet. Math 4653, Section 001 Elementary Probability Fall Ice Breaker Question: Do you prefer waffles or pancakes?

Transcription

1 Week 10 Worksheet Ice Breaker Question: Do you prefer waffles or pancakes? 1. Suppose X, Y have joint density f(x, y) = 12 7 (xy + y2 ) on 0 < x < 1, 0 < y < 1. (a) What are the marginal densities of X and Y? (b) What is the conditional density of Y given that X = 1/4? (c) Are X and Y independent? 1

2 2. Suppose a point is chosen uniformly at random from the triangle with vertices at (0,0), (1,0), and (0,1). (a) What are the marginal densities of X and Y? (b) What is the conditional density of X given that Y = 1/3? (c) What is the conditional density of X given that Y = 2/3? (d) Are X and Y independent? 2

3 3. Suppose X and Y have joint density f(x, y) = g(x)h(y). Let g(x) dx = c. (a) What is h(y) dy in terms of c? (b) What is f X (x) in terms of c? (c) What is f Y (y) in terms of c? (d) Prove X and Y are independent. 3

4 4. Show that if X = Binomial(n, p), Y = Binomial(m, p), and X, Y are independent, then X + Y = Binomial(n + m, p). 5. We say X = Gamma(n, λ) if X has density f X (x) = λn x n 1 (n 1)! e λx on 0 < x. Notice Gamma(1, λ) = Exponential(λ). Let X = Gamma(n, λ), Y = Exponential(λ), and X, Y be independent. Show X + Y = Gamma(n + 1, λ). 4

5 6. If X = Uniform(0, 5), Y = Uniform(3, 6), and X, Y are independent, then what is the density of X + Y? 7. Sujit must pass both a written test and a road test to get his driver s license. Each time he takes the written test he passes with probability 4/10, independently of other tests. Each time he takes the road test he passes with probability 7/10, also independently of other tests. What is the total number of expected tests Sujit must take before earning his license? 5

6 8. We draw 5 cards from a deck at once. What is the expected number of aces we get? 9. Suppose X and Y are independent. (a) Prove EXY = (EX)(EY ) if X, Y are discrete. (b) Prove EXY = (EX)(EY ) if X, Y are continuous. 6

7 10. Prove that var(x + Y ) = var(x) + 2cov(X, Y ) + var(y ). 11. An urn has 3 red balls and 2 green balls. We draw 2 balls from the urn and let X be the number of red balls we get. (a) If we draw without replacement, what is var(x)? (b) If we draw with replacement, what is var(x)? (c) In which case do we have more variability? 7

8 Real World Example: While the sign of the covariance of two random variables (or sets of data) tells us whether they tend to increase together or in opposition to each other, the covariance is other wise not a very useful metric. A large covariance might indicate that the variables vary together quite strongly, or it might just indicate that the variables take on large values. The more common tool for looking at how two random variables vary together is the Pearson product-moment correlation coefficient. This value, usually denoted ρ X,Y is computed as cov(x,y ) σ X σ Y, where σ X is the standard deviation of X and σ Y is the standard deviation of Y. This correlation coefficient can be between 1 and -1, and the closer ρ X,Y is to 1, the more correlated the variables. For more information: Correlation and dependence on Wikipedia. Correlation from The Research Methods Knowledge Base. 8