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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 Math 105 (Section 204) Probability 2011W T2 1/5 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 2011W T2 1/5 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 2011W T2 2/5 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 2011W T2 3/5 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 2011W T2 4/5 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 5/5