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Lecture

# UASTAT141Ch16-17.pdf

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University of Alberta

Statistics

STAT141

Paul Cartledge

Winter

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Ch. 16 – Random Variables
Def’n: A random variable is a numerical measurement of the outcome of a random
phenomenon.
A discrete random variableis a random variable that assumes separate values.
Æ # of people who think stats is dry
The probability distribution of a discrete random variable lists all possible values
that the random variable can assume and their corresponding probabilities.
Notation: X = random variable; x = particular value;
P(X = x) denotes probability that X equals the value x.
Ex16.1) Toss a coin 3 times. Let X be the number of heads.
8 possible values: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT
16abe0
x P(X = x)
0 0.125
1 0.375
2 0.375
3 0.125
Two noticeable characteristics for discrete probability distribution:
1. 0 ≤ P(X = x) ≤ 1 for each value of x
2. ∑ P(X = x) = 1
Ex16.2) Find the probabilities of the following events:
“no heads”: P(X = 0) = 0.125
“at least one head”: P(X ≥ 1) = P(X = 1) + P(X = 2) + P(X = 3)
= 0.375 + 0.375 + 0.125 = 0.875
“less than 2 heads”: P(X < 2) = P(X = 0) + P(X = 1) = 0.125 + 0.375 = 0.500
The population mean µ of a discrete random variable is a measure of the center of its
distribution. It can be seen as a long-run average under replication. More precisely,
µ = x P(X = x )
∑ i i
Sometimes referred to as µ = E(X) = the expected value of X.
Keep in mind that µ is not necessarily a “typical” value of X (it’s not the mode).
Ex16.3) Using Table 16X0,
µ = ∑ x i(X = x i = (0)(0.125) + (1)(0.375) + (2)(0.375) + (3)(0.125) = 1.5
Æ On average, the number of heads from 3 coin tosses is 1.5.
Ex16.4) Toss an unfair coin 3 times (hypothetical). Let X be as in previous example.
x P(X = x) µ = x P(X = x )
∑ i i
0 0.10 = (0)(0.10) + (1)(0.05) +
1 0.05 (2)(0.20) + (3)(0.65)
2 0.20 = 2.4
3 0.65 nd
As 2 example shows, interpretation of µ as a measure of center of a distribution is more
useful when the distribution is roughly symmetric, less useful when the distribution is
highly skewed.
The population standard deviation σ of a discrete random variable is a measure of
variability of its distribution. As before, the standard deviation is defined as the square
root of the population variance σ , given by
2 2 2 2
σ = ∑ (xi− µ) P(X = x )i= ∑ xiP(X = x )i− µ
Ex16.5) From Table 16X0,
2 2 222 2 2 2 1 1 3 3 4963
5 . 1 ] =) ( 3 ) ( 2 ) ( = σ− µ1 ∑ ( i i) [ 8 8 8 8 8484
Ch. 17 – Probability Models
Counting Techniques
A permutation of a set is an ordered sequence of the elements in the set.
n! = n × (n – 1) × (n – 2) × … × 2 × 1
Ex17.1) How many ways can the word COMEDY be arranged?
6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
If r ordered elements of a set of size n are desired, then
n n!
P r
(n− r)!
Ex17.2) How many ways can 3 of the letters in COMEDY be arranged?
P = 6! = 720 =120
3 (6 3)! 6
If order is NOT important when choosing r elements from a set of size n, then
⎛ ⎞ n!
C r ⎜ ⎟ =
⎝ ⎠ r n r )!
Ex17.3) How many ways can 3 letters be chosen from the word COMEDY?
⎛6 ⎞ 6! 720
⎜ ⎟= = = 20
⎝3 ⎠ 3!(6 3)! (6)(6)
The Binomial Distribution
Def’n: A Bernoulli trials a trial with only two possible outcomes, usually termed as a
“success” and a “failure”. The prob. of success is p and the prob. of failure is 1 – p.
Consider a random experiment consisting of n Bernoulli trials such that
1. The trials are independent.
2. Each trial results in only 2 possible outcomes: “success” and “failure”.
3. The probability of success in each trial (p) remains constant. Let an r.v. X be the number of successes in n trials where 0 < p < 1 and n = 1, 2, …
Then, an X with these parameters has a binomial distribution such that
⎛⎞ x nx
P())1== ( ⎜x − p x = 0, 1, …, n
⎝⎠
Also,
µ = E(X) = np and σ =V(X))= np(1− p
Ex17.4) Recall Oilers ticket phone line example from Ch. 15. Let X be the number of
calls succeeding in making a sale. Thus, X ~ B(n, p) = B(10, 0.3).
a) What is the probability that exactly 5 calls of 10 result in a sale?
⎛0 ⎞ 5 10−5
103 (X = 5) = ⎜ ⎟0.3 (1 0.3) = 0.
⎝5 ⎠
b) What is the probability that at least 3 calls result in a sale being made?
P(X )≥1−( ) b) or P(X ≤ a).
Ex17.5) Suppose we have a “uniform” distribution where obtaining each value

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