probability

Determine whether each of the following statement is true (i.e., always true) or false (i.e., not always true).

1. Let X be a random variable that takes values between 0 and c only, for some c≥0, so that P(0≤X≤c)=1. Then, var(X)≤c2/4. TRUE

2. X and Y are continuous random variables. If X∼N(μ,σ2) (i.e., normal with mean μ and variance σ2), Y=aX+b, and a>0, then Y∼N(aμ+b,aσ2). FALSE

3. The expected value of a non-negative continuous random variable X, which is defined by E[X]=∫∞0xfX(x)dx, also satisfies E[X]=∫∞0P(X>t)dt. TRUE

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