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Questions › JNU › JNU SSS 2013 › Q90
1 mark, −0.25Statistics › Random variables and standard distributions › Expectation and varianceModerateabout 1 min

JNU SSS 2013 Question 90: Random variables and standard distributions

X1,...,XNX_1, ..., X_NX1​,...,XN​ are independent and identically distributed random variables. Assume that XiX_iXi​ is normally distributed with mean 1 and variance 1. Let a1,...,aNa_1, ..., a_Na1​,...,aN​ be real numbers. Construct the random variable ZZZ as follows :

Z=∑i=1Nai×Xi2Z = \sum_{i=1}^{N} a_i \times X_i^2Z=∑i=1N​ai​×Xi2​

Which of the following is true?

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Plain-text version

X₁,..., X_N are independent and identically distributed random variables. Assume that Xᵢ is normally distributed with mean 1 and variance 1. Let a₁,..., a_N be real numbers. Construct the random variable Z as follows: Z = Σᵢ₌₁^N aᵢ × Xᵢ² Which of the following is true? (A) The expected value of Z is Σᵢ₌₁^N aᵢ (B) The expected value of Z is Σᵢ₌₁^N aᵢ² (C) The expected value of Z is 2 × Σᵢ₌₁^N aᵢ (D) None of the above

Also written as: X_1,..., X_N are independent and identically distributed random variables. Assume that X_i is normally distributed with mean 1 and variance 1. Let a_1,..., a_N be real numbers. Construct the random variable Z as follows: Z = sum_(i=1)^N a_i * X_i^2 Which of the following is true?

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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Question from the JNU SSS 2013 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.