1. Review of Chapter 4 - 5

2. Outline Random Variables Discrete Random Variable
Expected Value
Variance
Discrete Random Variable
Bernoulli and Binomial
Poisson
Continuous Random Variable
Normal Random Variables
Exponential Random Variables
Other continuous distributions*

3. Random Variables Real-valued functions defined on sample space.
The outcomes of coin flips
the function value is 1 if the outcome is H, and 0 if the outcome is T.
The number of heads in three flips of a coin.
Probability of random variables
Discrete random variables
Continuous random variables

4. S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}.
Probabilities of all possible values of X, if X is the number of heads in three coin flips.
S = {HHH, HHT, HTH, THH, HTT, THT, TTH,
TTT}.
P(X=0) = P{(T,T,T)} = 1/8
P(X=1) = P{(T,T,H), (T,H,T), (H,T,T)} = 3/8
P(X=2) = P{(T,H,H), (H,T,H), (H,H,T)} = 3/8
P(X=3) = P{(H,H,H)} = 1/8
If X is uniform in [0, 2], what is the probability of a particular value of X?
The frequency of a particular value in [0, 2] is 1/2.

5. Ex 1a. Suppose that X is a continuous random variable whose probability density function is given by
(a) What is the value of C?
(b) Find P(X > 1)

6. Expected Values
Expected value of random variable X is a weighted average of the possible values that X can take on, each value being weighted by the probability that X assumes it.
Discrete random variables
Continuous random variables
If X lies in between a and b and so its expected value.

7. Find E[X] where X is the outcome when we roll a fair die.
Since p(1) = p(2) = p(3) = p(4) = p(5) = p(6) = 1/6, we have
E[X] = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 7/2
Expected values can be out of the range of the possible values of the random variable.
Find E[X] when the density function of X is

8. Expectation of Function of Random Variables
Discrete random variables
Continuous random variables
E[aX+b] = aE[X] + b

9. Let X denote a random variable that takes on any of the values -1, 0, 1 with respective probabilities
P(X = -1) = .2, P(X = 0) = .5, P(X = 1) = .3
E[X2] = ?
E[X2] = (-1)2(.2) + 02(.5) + 12(.3) = .5
What is expected value of X2 if X is uniformly distributed over [0,1]?

10. Variance
Variance measures how far apart random variable X would be from its mean on the average.
Var(X) = E[(X-μ)2]
Var(X) = E[X2] – (E[X])2
Var(aX+b) = a2Var(X)
Compute variance of a random variable X
Find the expected value of X
Find the expected value of X2.

11. Cumulative Distribution Functions
Cumulative distribution function (cdf), also called distribution function, is defined on the real numbers by F(x) = P(X<=x).
Probability that X is smaller than certain value.
Every cdf is an increasing function.
Its limit at negative infinity (to the left) is 0 and its limit at positive infinity (to the right) is 1.
Discrete:
cdf is a step function.
Continuous

12. Distribution Function
Obtain distribution function from probability density function.
Discrete distribution
Differences between steps are the probability value at corresponding points
Continuous distribution
Integration

13. Compute Probabilities and PDF from Distribution Function - Discrete

15. Compute Probabilities and PDF from Distribution Function - Continuous
Let X be uniformly distributed over (0,1). What is the distribution function of the random variable Y, defined by Y = Xn?
Distribution function
Probability density function

16. Bernoulli and Binomial Random Variables
Experiments with two outcomes
H or T
success or failure
defective or qualified
Bernoulli (p): p(0) =1-p, p(1) = p, where p is the probability that the outcome is a success.
Binomial (n, p): n Bernoulli trials, where X is the number of successes that occur in the n trials.

17. A communication channel transmits the digits 0 and 1
A communication channel transmits the digits 0 and 1. However, due to static, the digit transmitted is incorrectly received with probability .2. Suppose that we want to transmit an important message consisting of one binary digit. To reduce the chance of error, we transmit instead of 0 and instead of 1. If the receiver of the message uses majority decoding, what is the probability that the message will be wrong when decoded?
The message is wrong if at least three errors made when transmitting 5 digits (at least three 0s when 1 is transmitted or at least three 1s when 0 is transmitted).
The number of errors made when transmitting 5 digits is Binomial with parameter (5, .2).

18. Poisson Random Variables
Poisson random variables are usually used in modeling rare events, where λ is the average number of occurrances of the event in certain interval.
Poisson random variable can be used as an approximation for a binomial random variable with parameters (n, p) when n is large and p is small so that np is a moderate size. λ = np.

19. Suppose that the probability that an item produced by a certain machine will be defective is .1. Find the probability that a sample of 10 items will contain at most 1 defective item.

20. The monthly worldwide average number of airplane crashes of commercial airlines is 3.5. What is the probability that there will be
(a) at least 2 such accidents in the next month
(b) at most 1 accidents in the next month?
Poisson with parameter λ = 3.5

21. Uniform Distribution
X is a uniform random variable on the interval (α, β) if its probability function is given by

22. Normal Distribution Bell-shaped & symmetrical
Random variable has infinite range
Mean measures the center of the distribution
Variance measures the spread of the distribution
 = Mean of random variable x 2 = Variance
 = …
e = …

23. Standard Normal
Normal distribution with parameter (0, 1), N(0,1), is also called standard normal.
If X is N(μ, σ2), then Z = (X - μ)/σ is standard normal.

24. For negative values of x:
For standard normal: P(Z ≤ -x) = P(Z > x)
For X ~ N(μ, σ)

25. If X is a normal random variable with parameter μ = 3 and σ2 = 9, find (a) P(2 < X < 5); (b) P(X > 0); (c) P(|X-3| > 6).

27. Normal Approximation to the Binomial Distribution
When n is large, a binomial random variable with parameter n and p will have approximately the same distribution as a normal random variable with the same mean and variance.
The ideal size of a first-year class at a particular college is 150 students. The college, knowing from past experience that on the average only 30 percent of those accepted for admission will actually attend, uses a policy of approving the applications of 450 students. Compute the probability that more than 150 first-year students attend this college.

28. Exponential Random Variables
The distribution of the amount of time until some specific event occurs.
The amount of time until an earthquake occurs.
Exponential random variables are memoryless.

29. Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked is an exponential random variable with parameter 1/20 (in thousand miles). Smith has a used car that he claims has been driven only 10,000 miles. If Jones purchases the car, what is the probability that she would get at least 20,000 additional miles out of it? Repeat under the assumption that the lifetime mileage of the car is not exponentially distributed but rather is uniformly distributed over (0, 40,000). How about the assumption is that the lifetime of the car is normally distributed with parameter (μ = 20,000, σ = 10,000)?
P(X>20) = 1 - P(X ≤ 20) = e-20*1/20 = e-1 ≈ 0.368
P(X > 30 | X > 10) = P(X>30) / P(X>10) = (1/4)/(3/4) = 1/3
P(X > 30 | X > 10) = P(X>30) / P(X>10) = P((X-20)/10)>1) / P((X-20)/10) > -1) = (Z > 1) / (Z > -1)