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**X**follows a certain distribution. Now say I multiply the random variable

**X**by a constant

*a*. Does the new random variable

*a*

**X**follow the same distribution as

**X**?

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- #2

EnumaElish

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Well, let's take it through the basics. If X ~ F_{X}, then F_{X}(x) = Prob{X < x} for __x__ < X < [itex]\bar x[/itex]. Let Y = aX for a__x__ < Y < a[itex]\bar x[/itex]; then F_{Y}(y) = Prob{Y < y} = Prob{aX < y} = Prob{X < y/a} = F_{X}(y/a). Therefore F_{Y}(y) = F_{X}(y/a) for a__x__ < Y < a[itex]\bar x[/itex].

Is F_{Y} the identical distribution as F_{X}? No. Does it belong to the same family as F_{X}? Yes, up to a scaling factor.

Is F

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EnumaElish

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The case of a = 0 is trivial, but you should bear that in mind, too.

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also X + d is normally distributed, for any real number d.

can anyone please show me the proof?thanks

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chiro

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If you want a systematic way to figure this out, use moment generating functions and if the structure of the mgf is the same as the unscaled distributions mgf, then you know that the distribution doesn't change and you can see how the parameters of the distribution have changed.Xfollows a certain distribution. Now say I multiply the random variableXby a constanta. Does the new random variableaXfollow the same distribution asX?

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chiro

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Like I said to the previous poster, use moment generating functions. Let U = cX + d. Use E[e^(tU)] be the mgf of U and you will find that this will give the mgf of a normal distribution with different parameters which will tell you what scaling and translating a normal does to its parameters. (Translating adds to the mean, scaling changes the variance by multiplying it by c^2).

also X + d is normally distributed, for any real number d.

can anyone please show me the proof?thanks

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