STAT 215
Review # 1
Fall 2016
Problem #1. Let X be a random variable with the following probability distribution
X
f(x)
-2
0.35
-1
0.25
3
0.30
4
0.10
(a) Calculate P ( X < 2), P(X ≥ 0)
(b) Find the mean E(X), variance and standard deviation of X.
(c) Construct the cumulative distribution function of X.
Problem #2. Let E be the event that a new car requires engine work under warranty and
let T be the event that the car requires transmission work under warranty. Suppose that
P(E) = 0.10, P(T)= 0.02, and P(E ∩T) = 0.01.
(a) Find the probability that the car needs work on either the engine, the transmission,
or both.
(b) Find the probability that neither the engine nor the transmission needs work.
(c) Find the probability that the car needs work on the engine but not on the
transmission.
Problem #3. A fair coin is tossed 10 times.
(a) What is the probability of obtaining exactly three heads?
(b) Find the mean number of heads obtained.
(c) Find the variance of the number of heads obtained.
(d) Find the standard deviation of the number of heads obtained.
Problem #4. A certain type of circuit board contains 300 diodes. Each diode has
probability p=0.002 of failing.
(a) What is the probability that exactly two diodes fail?
(b) What is the mean number of diodes that fail?
(c) What is the standard deviation of the number diodes that fail?
(d) A board works if none of its diodes fail. What is the probability that a board works?
Problem #5. In a lot of 10 microcircuits, 3 are defective. Four microcircuits are chosen
at random to be tested. Let X denote the number of tested circuits that are defective.
(a) Find P(X≤2)
(b) Find E(X)
(c) Find var(X)
Problem #6. A traffic light at a certain intersection is green 50% of the time, yellow
10% of the time, and red 40% of the time. A car approaches this intersection once each
day. Let X denote the number of days up to and including the third day on which a red
light is countered. Assume that each day represents an independent trial.
(a) Find P(X=7).
(b) Find E(X)
(c) Find var (X).