(i) E[X + Y ] = EX + EY . (ii) E[aX] = aEX
Practical use: If we can show that two random variables have the same PGF in Theorem 4.4: Let X be a discrete random variable with PGF GX(s). Then: 1. E ...
We prove the continuous case and leave Suppose that the random variables are discrete. We need to compute the expected value of the random variable E[X
10 mar 2006 The definition of expectation follows our intuition. Definition 1 Let X be a random variable and g be any function. 1. If X is discrete then ...
serve as the probability distribution for a discrete random variable X if and only if it s values Proof for case of finite values of X. Consider the case ...
Proposition 13.1 allows to show some of the properties of sums of independent random n] = m(n) (0). Example 13.10. Suppose X is a discrete random variable and ...
21 oct 2020 For a discrete random variable let x belong to the range of X. The ... Proof: If X and Y are simple random variables on the state space Ω ...
Recall that for a discrete random variable X we have a probability mass Show that X and Y are independent and find their joint density. Theorem 4. If X and Y ...
Calculating expectations for continuous and discrete random variables. 2. Conditional expectation: the expectation of a random variable X condi-.
Theorem: E(XY) = E(X)E(Y) when X is indepen- dent of Y. Proof: For discrete random variables X and Y
For a discrete random variable X we define the probability mass function (PMF) PROOF. Consider a random variable Z := X + Y which is a discrete random ...
Definition: Let X be a discrete random variable taking values in the non- Practical use: If we can show that two random variables have the same PGF in.
10 mars 2006 The definition of expectation follows our intuition. Definition 1 Let X be a random variable and g be any function. 1. If X is discrete then ...
Proof The fact that the range of X is either finite or countably infinite means Definition The expectation of a discrete random variable X is. E(X) =.
serve as the probability distribution for a discrete random variable X if and Proof for case of finite values of X. Consider the case where the random ...
Theorem: E(XY) = E(X)E(Y) when X is indepen- dent of Y. Proof: For discrete random variables X and Y
serve as the probability distribution for a discrete random variable X if and Proof for case of finite values of X. Consider the case where the random ...
The moment generating function (MGF) of a random variable X is a function mX In the discrete case mX is equal to ? x etxp(x) and in the continuous case.
The probability function of a discrete random variable x is given by p(x) = kx2 x = 1 2
Best estimate under 1-1(X=x) loss: mode. Ie choosing mode maximizes probability of being exactly right. Proof easy for discrete r.v.'s; a limiting argument is
The set of possible values X can take on is its range/support denoted X If Xis nite or countable in nite (typically integers or a subset) X is a discrete random variable (drv) Else if Xis uncountably large (the size of real numbers) X is a continuous random variable Example(s) Below are some descriptions of random variables
Theorem 3 4 1: Variance Adds for Independent RVs If X ?Y then Var(X + Y) = Var(X) + Var(Y) This will be proved a bit later but we can start using this fact now! It is important to remember that you cannot use this formula if the random variables are not independent (unlike linearity)
use the fact that X is a sum of n independent Bernoulli variables Because the Bernoulli variables have expectation p E[X] = np Because they have variance p(1?p) Var(X) = np(1?p) 4 Geometric random variables Suppose we keep trying independent Bernoulli variables until we have a success; each has probability of success p
Then X is a discrete random variable with a geometric distribution: X ~ G or X ~ G (0.0128). What is the probability of that you ask ten people before one says he or she has pancreatic cancer?
Chapter 4 Discrete Probability Distributions Chapter 4 Discrete Probability Distributions 4.1 Random variable A random variable is a function that assigns values to di?erent events in a sample space. Example 4.1.1. Consider the experiment of rolling two dice to- gether.
2. X is a continuous random variable taking values between 1 and 2. If PCX 1.3) pl and 2. Xis a continuous random variable taking values between 1 and 2. If PCX S 1.3)-p27 C pl< p2 C pl>p2 C Not enough information p1 = p2 3. X is a continuous Question: 1. X is a discrete random variable that takes values (1,2, 3, 4, 5).
f(y)dy ?f(x) ·dx. ?We can characterize the distribution of a continuous random variable in terms of its 1.Probability Density Function (pdf) 2.Cumulative Distribution Function (cdf) 3.Moment Generating Function (mgf, Chapter 7) ?Theorem. If f is a pdf, then there must exist a continuous random variable with pdf f. PX({X = x})=