ROCHESTER ECE 270 - Handout 5 - Introduction to Random Variables

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ECE270: Handout 5Introduction to Random Variables (RVs)Outline:1. informal definition of a RV,2. three types of a RV: a discrete RV, a continuous RV, and a mixed RV,3. a general rule to find probability of events concerning a RV,4. cumulative distribution function (CDF) of a RV,5. formal definition of a RV using CDF,6. discrete RV: probability mass function (pmf) and CDF,7. continuous RV: probability density function (pdf ) and CDF,8. basic properties of the CDF.• The outcome of a random experiment need not be a numbers. Examples are: coding theincoming patients in a hospital according to their insurance and health status, randomlyselecting a committee from a group of people, randomly selecting balls from an urn, randomlyselecting cards from a deck.• Usually we are interested in some measurement or numerical attribute of the outcome. Ex-amples are: counting the number of heads when tossing a coin 10 times, the number ofre-transmission needed until the receiver receives the data packet correctly, the number of er-rors in erroneous received data packets, the lifetime of a memory chip, the number of packetsarriving in t sec at a server, the number of queries arriving in t sec at a call center, the numberof particles emitted by a radioactive mass during a fixed time period, the random thermalnoise being added to the signal at the receiver of a communication system at a specific time.• In these examples, we assign a real number to the outcome of the random experiment throughmeasurement. Since the outcomes are random, the results of the measurements will be randomtoo. Hence it makes sense to talk ab out the probabilities of the resulting numerical values.F Informal Definition of a RV• A RV X is a function that assigns a real valued number x = X (ξ) to each outcome ξ ∈ S(Recall: a function is a rule for assigning a numerical value to each element of a set).• Sample space S is the domain of the RV X and the set of all real numbers taken on by X,Sx, Sx⊂ R, is the range of the RV.F Three Types of a RV• Three types of RVs: i) discrete, ii) continuous, iii) mixed.i) The range of a discrete RV X is a countable set (either finite or infinite) Sx={x1, x2, ...} or Sx= {x1, x2, ..., xn}.ii) The range of a continuous RV X is an uncountable set, e.g., Sx= [0, ∞), Sx=(−∞, ∞), Sx= [0, 1], Sx= [a, b] for −∞ < a < b < ∞.iii) The range of a mixed RV X is the union of an uncountable and a countable sets.• Notation: capital letters X, Y , Z, V , U , ... denote RVs, lowercase letters x, y, z, v, u, ... de-note possible values of RVs.• Consider the RV X, the function or rule that assigns a real number x = X(ξ) to the outcomeξ ∈ S is fixed and deterministic, e.g., the rule of “count the number of heads when we toss acoin 3 times”. The randomness in the experiment is complete as soon as we toss the coin 3times. The process of counting is deterministic.• In some random experiments the outcome ξ is already the numerical value we are interested in,e.g., measure the lifetime of a chip under certain conditions, ⇒ X(ξ) = ξ (identity function).• The distribution of the values of a RV X is determined by the probabilities of the basic eventsof the underlying random experiment, i.e., we should be able to compute the probability ofthe observed value of X in terms of the probability of the underlying event.• The specification of the measurements on the outcomes of a random experiment defines afunction on S and hence a RV.• A function of a RV is another RV.EXAMPLE 1A fair coin is tossed 3 times and the sequence of heads and tails is noted. The sample spaceis S = {HHH, HHT, HT H, HT T, T HH, T HT, T T H, T T T } and the outcomes are equallyprobable.a) Let X be the number of heads in the 3 tosses. Sx= {0, 1, 2, 3}.b) A player pays 1.5$ to play the following game: the player receives 1$ if X = 2 and8$ if X = 3, but nothing otherwise. Let Y be the reward to the player. Sy= {0, 1, 8}.c) Let Z be a function of X such that Z = 0 if X ∈ {0, 1, 2} and Z = 1 if X = 3.Sz= {0, 1}.ξ HHH HHT HTH THH HTT THT TTH TTTX(ξ) 3 2 2 2 1 1 1 0Y (ξ) 8 1 1 1 0 0 0 0Z(ξ) 1 0 0 0 0 0 0 0P (X = 0) = P ({T T T }) = 1/8P (X = 1) = P ({T T H, T HT, HT T }) = 3/8P (X = 2) = P ({T HH, HHT, HT H}) = 3/8Page 2P (X = 3) = P ({HHH}) = 1/8P (Y = 0) = P ({T T T, T T H, T HT, HT T }) = 4/8 = 1/2P (Y = 1) = P ({T HH, HT H, HHT }) = 3/8P (Y = 8) = P ({HHH}) = 1/8½P (Z = 0) = P (X ∈ {0, 1, 2}) = P (X = 0) + P (X = 1) + P (X = 2) = 1/8 + 3/8 + 3/8 = 7/8P (Z = 1) = P (X = 3) = 1/8or alternatively:½P (Z = 0) = P ({T T T, T T H, T HT, HT T, HHT, HT H, T HH}) = 7/8P (Z = 1) = P ({HHH}) = 1/8F A General Rule to Find Probabilities of Events Concerning a RV X• The example shows a general technique to find the probability of events involving RVs. Tofind the probability of X ∈ B (where B ⊂ R) we need to find the set of outcomes A, A ⊂ Sthat are mapped to B, i.e., the set A = {ξ : X(ξ) ∈ B}.• If the experiment outcome ξ ∈ A then event A occurs. Hence X(ξ) ∈ B ⇒ event B occurs.If event B occurs then X (ξ) ∈ B implies ξ ∈ A ⇒ event A occurs. So: P (X ∈ B) = P (A) =P ({ξ : X (ξ) ∈ B}). We refer to A and B as equivalent events.• In some random experiments the outcome ξ is already the numerical value we are interestedin. In such cases we simply let X(ξ) = ξ, i.e., the identity function is used to obtain a randomvariable. Example: measure the received signal at a receive antenna.F CDF of a RV X• Regardless of X being discrete, continuous, or mixed, the cumulative distribution function(CDF) of a RV X is defined as the probability of the event B = {X ≤ x}:FX(x) = P (B) = P ({X ≤ x}) , P (X ≤ x) for − ∞ < x < ∞The event B = {X ≤ x} and its probability vary as x is varied. Hence, FX(x) is a functionof the variable x.• In terms of the underlying random experiment FX(x) = P ({ξ : X (ξ) ≤ x}| {z }an event).F …


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