# Math 419 – Spring 2015 Joint Distribution(Continuous RV) Problem Set 8a X

```Math 419 – Spring 2015
Joint Distribution(Continuous RV)
Problem Set 8a
1. Let X and Y be continuous random variables with joint density function f(x, y) = (3/4)x, for 0 &lt; x &lt; 2
and for 0 &lt; y &lt; 2 – x, and zero otherwise. Calculate P(X &gt; 1).
2. Let T1 be an exponential random variable with mean 15 and T2 be an exponential random variable
with mean 30. Assume T1 and T2 are independent. Find the probability that the maximum of T1 and
T2 is less than 20.
3. Let X and Y be continuous random variables with joint density function f( x, y) = x + y, for
0 &lt; x &lt; 1 and 0 &lt; y &lt; 1, and zero otherwise. What is the marginal density of X ?
4. Suppose the joint cumulative distribution function of X and Y is F(x, y) = xy(x2 + y2 – xy), for
0 &lt; x &lt; 1 and 0 &lt; y &lt; 1. What is the variance of Y ?
5. Smith and Jones' future lifetimes are independent and distributed uniformly between 5 to 20 years.
What is the probability that they die within a year of each other?
6. Smith has 50,000 in a money market account, whose annual effective interest rate X is uniformly
distributed on [0.03, 0.05] and 30,000 in stock market, whose annual effective interest rate Y is
uniformly distributed on [– 0.05, 0.1]. Smith knows that X and Y are independent. What is the
probability that the interest accrued on his investments exceeds 3,000 in the next year?
7. A car crosses an intersection between 7:15am and 7:30am each day, uniformly distributed. A train
holds up traffic for five minutes starting sometime between 7:20am and 7:30am, also uniformly
distributed. What is the probability that the waiting time of the car is less than 3 minutes?
8. Find the cumulative distribution function for the sum of two independent exponential random
variables with means 5 and 7.
9. The joint probability density function for X and Y is f(x, y) = 8xy, 0 &lt; x &lt; y &lt; 1. Calculate
E[Y|X = 1/2]
```