QUESTION 17 Let X1, X2, …, Xn be IID random variables, where...

QUESTION 17 Let $X_1, X_2, \dots, X_n$ be IID random variables, where $X_i \sim Exponential(\lambda)$. The upper bound to $Pr[X_1 + X_2 + \dots + X_n \ge kn]$ given by Markov's inequality is $\circ \frac{n\lambda}{k}$ $\circ \frac{\lambda}{nk}$ $\circ \frac{n}{k\lambda}$ $\circ \frac{1}{k\lambda}$

Transcript

00:01 Hello students in this question it is asked to to probability of x greater than or equal to a is less than or equal to sigma square divided by sigma square plus a square and also it is given expectation of x is equal to zero and expectation of x square will be equal to sigma square that is variance of x is sigma square since here expectation of x is 0, expectation of x square will be sigma square because variance of x means expectation of x square minus expectation of x whole square and this is equal to zero.

00:47 So expectation of x square will be equal to sigma square.

00:52 So first let us consider x greater than or equal to a.

01:00 That implies x plus c is greater than or equal to a plus c so x plus c whole square is greater than or equal to a plus c whole square so x plus c whole square is greater than or equal to a plus c whole square which implies mod of x plus c is greater than or equal to a plus c whole square which implies mod of x plus c is greater than or equal to a plus c that means x plus c is greater than a plus c or x plus c is less than minus a plus c so now consider probability of x greater than or equal to a is less than or equal to probability of x plus c whole square greater than or equal to a plus c whole square so this can be written as less than or equal to expectation of x plus c whole square divided by a plus c whole square and this is by marco inequality for all c greater than zero and a greater than zero so now define g of c such that expectation of x plus c whole square divided by a plus c whole square that equal to g of c so this can be written as expectation of x square plus c square because it is a constant divided by a plus c whole square so this is equal to sigma square plus c squared divided by a plus c whole square so now differentiating that is g dash of c equals to 2c divided by a plus c whole square minus 2 into here it is a plus c cube and here sigma square plus c square.

03:32 So here g dash of c is greater than or equal to 0 implies c is greater than or equal to sigma square divided by a...

Question

QUESTION 17 Let $X_1, X_2, \dots, X_n$ be IID random variables, where $X_i \sim Exponential(\lambda)$. The upper bound to $Pr[X_1 + X_2 + \dots + X_n \ge kn]$ given by Markov's inequality is $\circ \frac{n\lambda}{k}$ $\circ \frac{\lambda}{nk}$ $\circ \frac{n}{k\lambda}$ $\circ \frac{1}{k\lambda}$

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