(a) Verify that fx,y is indeed a probability density function. (b) Find the marginal probability density function fx and state the name of the distribution of X. (c) Find the conditional probability density function fy|x=x.

The probability density function

Let X be a random variable with the probability density function f(x) = { ae; x>0 0 ; otherwise Find (i) Value of a

The random-number generator on calculators randomly generates a number between 0 and 1. The random variable X, the number generated, follows a uniform probability distribution. (a) Identify the graph of the uniform density function. (b) What is the probability of generating a number between 0.48 and 0.62? (c) What is the probability of generating a number greater than 0.85? (a) Choose the correct graph of the uniform density function below. O A. O B. Density 1.2- 1- 0.8- 6. 0.6- 0 0.2 0.4 0.6 0.8 11.2 0.4- 0.2- to (b) The probability is (c) The probability is Q M (Simplify your answer.) (Simplify your answer.) ADensity 1.2- 1- 0.8- 0.6- 0.4- 0.2- to 0 0.2 0.4 0.6 0.8 1 1.2 Q M O C. 1.2- 1- 0.8- 0.6- 0.4- 0.2- Density 0+ 0 0.2 0.4 0.6 0.8 1 12 Q Q

Find the value of k that makes the given function a probability density function on the specified interval. f(x) = kx 2 ≤ x ≤ 6 k= || =

Verify Property 2 of the definition of a probability density function over the given interval. 1 10 f(x)= [4,6]