10_DiscreteRVs-2_Expectation_packed

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Unformatted text preview: Defini:ons [ var( X ) = E ( X − E [ X ]) 2 Ilya Pollak Calcula:ng var(X) var( X ) = E [( X − E [ X ]) ] = ∑ ( x − E [ X ]) p ( x ) 2 2 X x = ∑ ( x 2 − 2 xE [ X ] + ( E [ X ]) 2 ) pX ( x ) x = ∑ x 2 pX ( x ) − 2∑ xE [ X ] pX ( x ) + ∑ ( E [ X ]) 2 pX ( x ) x x x = ∑ x 2 p X ( x ) −2 E[ X ] ∑ xp X ( x ) + ( E[ X ])2 ∑ p X ( x ) x x x E[ X ] 1 E[ X 2 ] − ( E [ X ])2 = E [ X 2 ] − ( E [ X ]) 2 [ Sometimes this is easier to compute than E ( X − E [ X ]) 2 Ilya Pollak Example 2.4: Variance of a Bernoulli random variable Ilya Pollak Another Bernoulli random variable Toss a coin with P (H) = p, and let ⎧ a if H Y =⎨ ⎩ b if T ⎧ p if k = a Then pY ( k ) = ⎨ ⎩1 − p if k = b Note : Y − b ⎧1 if H =⎨ =X a − b ⎩0 if T How to compute the expectation and variance of this random variable? Therefore, Y = X ( a − b) + b E [Y ] = E [ X ]( a − b) + b = p( a − b) + b E [Y 2 ] = E [ X 2 ( a − b) 2 + 2 X ( a − b)b + b 2 ] = E [ X 2 ]( a − b) 2 + 2 E [ X ]( a − b)b + b 2 = p( a − b) 2 + 2 p( a − b)b + b 2 Ilya Pollak Another Bernoulli random variable Toss a coin with P (H) = p, and let ⎧ a if H Y =⎨ ⎩ b if T ⎧ p if k = a Then pY ( k ) = ⎨ ⎩1 − p if k = b Note : Y − b ⎧1 if H =⎨ =X a − b ⎩0 if T How to compute the expectation and variance of this random variable? Therefore, Y = X ( a − b) + b E [Y ] = E [ X ]( a − b) + b = p( a − b) + b E [Y 2 ] = p( a − b) 2 + 2 p( a − b)b + b 2 var(Y ) = E[Y 2 ] − ( E[Y ])2 = ( p − p 2 )(a − b )2 Ilya Pollak Standard devia:on as measurement error •  Run a series of independent, iden:cal experiments, e.g., Bernoulli trials. •  Empirically es:mate the probability of an event, say, the success in a Bernoulli trial, as the number of successes divided by the number of experiments. •  Standard devia:on characterizes by how much we would expect our es:mate to deviate from the actual probability of success, over many experiments. •  For example, if p=0.2, then E[X] = 0.2, var(X)=0.16, and standard devia:on of X is 0.4. Ilya Pollak Root mean- square devia:on of the es:mate of p from 0.2, as a func:on of the number of trials Ilya Pollak Standard devia:on as risk •  Suppose you have two investment opportuni:es: –  Opportunity 1: invest $1000, have equal chances of a total wipeout or of $2000 profit –  Opportunity 2: invest $1000, earn $500 profit guaranteed. •  Expected profit for 1 is 0.5(−$1000) + 0.5($2000) = $500. •  Expected profit for 2 is $500. •  But clearly the two opportuni:es are very different: you risk a lot under the first one...
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This note was uploaded on 09/11/2013 for the course ECE 302 taught by Professor Gelfand during the Fall '08 term at Purdue University-West Lafayette.

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