Download Chapter 5 Random variable, Binomial distribution and other discrete

Chapter 5 Random variable, Binomial distribution and other discrete distributions.
Some challenging questions on random variables:
1. Suppose the random variable X takes only two values with P (X = 1) = p, and P (X = 0 = 1 − p. Find the mean,
variance and standard deviation of X. For what values of p is the variance of X greatest?
2. Suppose X is a discrete random variable with mean µx , and variance σx2 . Let Y be a linear function of X, such that
Y = aX + b, where a, and b are constants. Find the mean and variance of Y in terms of µx and σx2 .
3. Variance computational Formula: Prove that Var(X) = E(X 2 ) − µ2 .
4. Standardization: Suppose X is a discrete random variable with mean µx and variance σx2 . Let Y be defined in terms
of X by
X − µx
.
Y =
σx
Find the mean, variance and standard deviation of Y.
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Example: Binomial experiment with n=5.
Consider a binomial experiment with n = 5 trials and probability of success p.
a) A typical outcome with two successes is SFFSF. Find the probability of this outcome.
P (SF F SF ) = p2 (1 − p)3 .
b) Another possible coutcome with two successes is FSFSF. Find the probability of this outcome.
P (F SF SF ) = p2 (1 − p)3 .
c) Compare your results from (a) and (b).
It is observed that the two probabilities are identical.
d)Find the probability that X = 2 successes.
P (X = 2) = number of outcomes with 2 successesp2 (1 − p)3 =5 C2 p2 (1 − p)3 .
The general formulae:
If X ∼ B(n, p), then P (X = x) =n Cx px (1 − p)n−x .
Practice: 1) Seventy-five percent of all ice wines in Canada is made in Ontario. Suppose 10 bottles of Canadian ice
wine are selected at random.
a) Find the probability that exactly six bottles are from Ontario.
b) Find the probability that at least seven bottles are from Ontario.
Next we introduce the concept of cumulative probability:
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P (X ≤ x) =
Px
k=0
P (X = k) = P (X = 0) + P (X = 1) + P (X = 2) + · · · + P (X = x).
We will use the cumulative probability function in R or cumulative probability table at the back of our textbook to
solve the following question.
Practice:
1) Approximately 40% of all pizza orders are carry-out. Suppose 20 pizza orders are randomly selected.
a. Find the probability that at most 8 are carry-out orders.
b. Find the probability that exactly 10 are carry-out orders.
c. Find the probability that at least 7 are carry-out orders.
d. Find the probability that between 5 and 11 (inclusive) are carry-out orders.
The R command is: pbinom(8,20,0.4)
Answer is 0.5955987.
The mean and variance formula for Binomial distribution:
If X ∼ B(n, p), then E(X) = np, and Var(X) = np(1 − p). Can you prove this result?
Example:
According to data, 60% of all children in Detroit live in poverty. Suppose 100 children in Detroit are selected at
random.
a) Find the mean, variance, and standard deviation of the number of children living in poverty.
b) Suppose 55 of the 100 children are living in poverty. Is there any evidence to suggest the reported percentage is
false?
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Section 5.5 Other discrete distribution
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