p x k

Thus the expression P(X = x) symbolizes the probability that the random variable X takes on the particular value x Often this is written simply as P(x) Likewise P(X ≤ x) = probability that the random variable X is less than or equal to the specific value x; P(a ≤ X ≤ b) = probability that X lies between values a and b

Z(z) = P(Z z) = P(g(X;Y) z) = P(f(x;y) : g(x;y) zg) = Z Z Az p X;Y(x;y)dxdy: 3 The pdf is p Z(z) = F0 Z (z) Example 5 Practice problem Let (X;Y) be uniform on the unit square Let Z= X=Y Find the density of Z 5 Important Distributions Normal (Gaussian) X˘N( ;˙2) if p(x) = 1 ˙ p 2ˇ e (x )2=(2˙2: If X2Rd then X˘N( ;) if p(x) = 1 (2ˇ

1 Bernoulli distribution with success probability p: With 0 < p < 1 a constant X has p m f p(k) = P(X = k) given by p(1) = p p(0) = 1−p p(k) = 0 otherwise Thus X only takes on the values 1 (success) or 0 (failure) A simple computation yields E(X) = p Var(X) = p(1−p) M(s) = pes +1−p

1 Probability Density Functions (PDF) For a continuous RV X with PDF f X (x) b 6 041/6 431 Probabilistic Systems P (a ≤ X ≤ b)= f X (x)dx a Analysis P (X ∈ A)= f X (x)dx A Quiz II Review Properties: Fall 2010 • Nonnegativity: f X (x) ≥ 0 ∀x • Normalization: ∞ f X (x)dx =1 −∞ 1 2 3 Mean and variance of a

Definition Suppose a discrete random variable X has the following pmf P(X = k) = qkP; 0 k < 1 The X is said to have geometric distribution with parameter P Remark Usually this is developed by replacing “having a child” by a Bernoulli experiment and having a girl by a “success” (PC) I could have used coin flips Proposition