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Questions › GATE › GATE 2021 › Q40
2 marksStatistics › Random variables and standard distributions › Expectation and varianceModerateabout 2 min

GATE 2021 Question 40: Random variables and standard distributions

Consider that a sample of size 3 is randomly drawn from a population that takes only two values, equally likely: −1-1−1 and 111. Let z=max⁡(x1,x2,x3)z=\max(x_1,x_2,x_3)z=max(x1​,x2​,x3​) where x1,x2,x3x_1,x_2,x_3x1​,x2​,x3​ are the sample observations. The expected value of zzz, E(z)E(z)E(z) is ______ (round off to two decimal places).

Answer and worked solution

Answer: 0.75

Each xix_ixi​ independently takes the value −1-1−1 or 111, each with probability 12\dfrac1221​. There are 23=82^3=823=8 equally likely outcomes for (x1,x2,x3)(x_1,x_2,x_3)(x1​,x2​,x3​).

z=max⁡(x1,x2,x3)z=\max(x_1,x_2,x_3)z=max(x1​,x2​,x3​) equals −1-1−1 only in the single outcome where all three draws are −1-1−1 (since if even one draw is +1+1+1, the max becomes +1+1+1). This happens with probability:

P(z=−1)=P(x1=−1)×P(x2=−1)×P(x3=−1)=(12)3=18P(z=-1) = P(x_1=-1)\times P(x_2=-1)\times P(x_3=-1) = \left(\frac12\right)^3 = \frac18P(z=−1)=P(x1​=−1)×P(x2​=−1)×P(x3​=−1)=(21​)3=81​

In all other cases (probability 1−18=781-\dfrac18=\dfrac781−81​=87​), at least one draw is +1+1+1, so z=+1z=+1z=+1.

E(z)=(−1)×18+(1)×78=−18+78=68=34=0.75E(z) = (-1)\times\frac18 + (1)\times\frac78 = -\frac18+\frac78 = \frac68 = \frac34 = 0.75E(z)=(−1)×81​+(1)×87​=−81​+87​=86​=43​=0.75

Therefore E(z)=E(z) = E(z)= 0.75.

The trap: 0.00

Students sometimes assume by symmetry that E(max⁡)E(\max)E(max) should simply equal the average of the two possible values ((−1+1)/2=0(-1+1)/2=0(−1+1)/2=0), applying the logic that would be correct for E(xi)E(x_i)E(xi​) itself, but not for the max of multiple draws — the max operator is not symmetric/linear in this way, and correctly computing E(z)E(z)E(z) requires working out the actual probability distribution of zzz first.

Plain-text version

Consider that a sample of size 3 is randomly drawn from a population that takes only two values, equally likely: -1 and 1. Let z=max (x₁,x₂,x₃) where x₁,x₂,x₃ are the sample observations. The expected value of z, E(z) is (round off to two decimal places).

Also written as: Consider that a sample of size 3 is randomly drawn from a population that takes only two values, equally likely: -1 and 1. Let z=max (x_1,x_2,x_3) where x_1,x_2,x_3 are the sample observations. The expected value of z, E(z) is (round off to two decimal places).

Random variables and standard distributions has come up 173 times in 7 exams since 2005.

What this question tests

This question is on Expectation and variance, part of Random variables and standard distributions in Statistics.

The mean and spread of a random variable, and how they change under transformations and sums.

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