Computing probabilities
Suppose that a pair of fair dice are rolled simultaneously. What is the
probability that the product of the two face values is odd?
A. 1/36
B. 1/4
C. 1/6
D. 3/4
Math 105 (Section 204)
Probability
2011W T2
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Computing probabilities
Suppose that a pair of fair dice are rolled simultaneously. What is the
probability that the product of the two face values is odd?
A. 1/36
B. 1/4
C. 1/6
D. 3/4
Questions to think about: What is the probability density function of
the product of face values? Graph the cumulative distribution function.
Math 105 (Section 204)
Probability
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Working with a PDF
What is the value of k so that
f (x) = kx(1 − x)4 ,
0≤x ≤1
is a probability density function?
A. 1
B. 5
C. 30
D. 6
Math 105 (Section 204)
Probability
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The normal distribution
A continuous random variable has a pdf given by
(∗)
f (x) = √
If you are given that
Z
1
2πσ 2
a
e−
(x−µ)2
2σ 2
,
−∞ < x < ∞.
z2
e − 2 dz = p,
−∞
find Pr(µ − aσ < X < µ + aσ) in terms of p.
A. 1 − 2p
B. 1 − p
C. p
D. 2p − 1
Remark: A random variable whose pdf is of the form (∗) is called a
normal random variable with mean µ and standard deviation σ.
Math 105 (Section 204)
Probability
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Expectation and Variance
Discrete case
If X is a discrete random variable taking values x1 , x2 , · · · , xn with
probabilities p! , p2 , · · · , pn respectively, then
E(X ) = µ =
n
X
xk pk ,
k=1
n
X
Var(X ) = σ 2 =
(xk − µ)2 pk .
k=1
Math 105 (Section 204)
Probability
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Expectation and Variance (ctd)
Continuous case
If X is a continuous random variable with PDF f , then
Z ∞
E(X ) = µ =
xf (x) dx,
−∞
Z ∞
Var(X ) = σ 2 =
(x − µ)2 f (x) dx.
−∞
µ measures the “mean” or “average value” of the random variable, while
σ (the standard deviation) measures the spread.
Math 105 (Section 204)
Probability
2011W T2
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