STAT 350 (Spring 2019)
Homework 4 (26 points + 1 point BONUS)
1
Practice Problems: 5.1 (p.191), 5.5 (p.191), 5.7 (p.191),
Practice Problems: 5.21 (p.200), 5.45 (p.208), 5.47 (p.208)
randomized , autograded
(1 point) 1.
Classify each random variable as discrete or continuous.
Please see LON CAPA for the correct answers.
(1 point) 2.
Show that Var(X) =
σ
2
= E(X
2
) − (
E(X))
2
. Hint: Use the definition of variance and
μ
= E(X).
Var(?) = ? [(? − ?(?))
2
] by definition of variance
= ?[(? − ?)
2
] as ? = ?(?)
= ?[?
2
− 2?? + ?
2
] by multiplication
= ?(?
2
) − 2??(?) + ?
2
by properties of ? and ? being a constant
= ?(?
2
) − 2(?(?))
2
+ (?(?))
2
as ? = ?(?)
= ?(?
2
) − (?(?))
2
algebra
Practice Problems: 5.21 (p.200), 5.45 (p.208), 5.47 (p.208)
Additional Problems: 5.33 (p.201), 5.39abc (p.201), 5.57 (p.209), 5.59 (p.209)
(1 point) 3.
BONUS: Show using the definition of expected values for discrete random variables
(Eq. 5.1) that for a discrete random variable, X, E(g(X
)) = ∑
g(x) p(x) where g(x) is linear. That
is, show the formula is true assuming that g(x)
= ax + b. You are showing that a special case
of Eq. 5.2 is true; therefore, you many not use that formula.
?[?(?)] = ?[?? + ?] by the definition of ?
= ? ?[?] + ? by properties of expectation
= ? ∑ 𝑥 𝑝(𝑥)
𝑥∈?
+ ? by the definition of expectation for a discrete random variable
= ? ∑ 𝑥 𝑝(𝑥)
𝑥∈?
+ ? ∑ 𝑝(𝑥)
𝑥∈?
as all probabilities sum to 1
= ∑ ? 𝑥 𝑝(𝑥)
𝑥∈?
+ ∑ ? 𝑝(𝑥)
𝑥∈?
= ∑(? 𝑥 𝑝(𝑥) + ? 𝑝(𝑥))
𝑥∈?
by properties of summation
= ∑(?𝑥 + ?)𝑝(𝑥)
𝑥∈?
= ∑ ?(𝑥)𝑝(𝑥)
𝑥∈?
.
Practice Problems: 5.21 (p.200), 5.45 (p.208), 5.47 (p.208)
Additional Problems: 5.33 (p.201), 5.39abc (p.201), 5.57 (p.209), 5.59 (p.209)

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