3) the length of time to failure (in hundreds of hours) for a transistor is a random variable y with distribution function give by f(y) = 0, y < 0 1 − e −y 2 , y ≥ 0 a) show that f(y) has the properties of a distribution function. b) find f(y). c) find the probability that the transistor operates at least 200 hours.

Looks like

a. is a valid distribution function if

The first two conditions are met: we have

The second condition is also met:

both of which are non-negative over their respective domains.

Finally, the fourth condition is met because is continuous everywhere.

b. We already found above when checking for the monotonicity of ; the probability density is simply the derivative of the distribution function .

(Notice that we include so that the exists.)

c. By definition of the distribution function, we have

which is value very close to 0.

paula, cory, and krista