1. Example Suppose X ~ Uniform(2, 4). Let . Find .
What if X ~ Uniform(-4, 4)?
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2. Functions of Random variables
In some case we would like to find the distribution of Y = h(X) when the distribution of X is known.
Discrete case
Examples
1. Let Y = aX + b , a ≠ 0
2. Let
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3. Continuous case – Examples
1. Suppose X ~ Uniform(0, 1). Let , then the cdf of Y can be found as
follows
The density of Y is then given by
2. Let X have the exponential distribution with parameter λ. Find the
density for
3. Suppose X is a random variable with density
Check if this is a valid density and find the density of
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4. Question
Can we formulate a general rule for densities so that we don’t have to look at cdf?
Answer: sometimes …
Suppose Y = h(X) then
and
but need h to be monotone on region where density for X is non-zero.
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5. Check with previous examples: 1. X ~ Uniform(0, 1) and
2. X ~ Exponential(λ). Let
3. X is a random variable with density
and
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6. Theorem
If X is a continuous random variable with density fX(x) and h is strictly increasing and differentiable function form R  R then Y = h(X) has density
for
Proof:
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7. Theorem
If X is a continuous random variable with density fX(x) and h is strictly decreasing and differentiable function form R  R then Y = h(X) has density
for
Proof:
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8. Summary If Y = h(X) and h is monotone then Example X has a density
Let Compute the density of Y.
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9. Exercise Find the density of X = Z2 where Z ~ N(0,1).
This is the Chi-Square density with parameter 1. Notation:
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10. Change-of-Variable for Joint Distributions
Theorem
Let X and Y be jointly continuous random variables with joint density
function fX,Y(x,y) and let DXY = {(x,y): fX,Y(x,y) >0}. If the mapping T given
by T(x,y) = (u(x,y),v(x,y)) maps DXY onto DUV. Then U, V are jointly
continuous random variable with joint density function given by
where J(u,v) is the Jacobian of T-1 given by
assuming derivatives exists and are continuous at all points in DUV .
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11. Example Let X, Y have joint density function given by
Find the density function of
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12. Examples
Suppose X~ Poisson(λ1) independent of Y~ Poisson(λ2). Find the
distribution of X+Y.
X, Y independent each having Exponential distribution with mean 1/λ. Find the density for W=X+Y.
If X, Y independent standard normal random variables, find the density of W=X+Y.
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13. Density of Quotient
Suppose X, Y are independent continuous random variables and we are interested in the density of
Can define the following transformation
The inverse transformation is x = w, y = wz. The Jacobian of the inverse transformation is given by
Apply 2-D change-of-variable theorem for densities to get
The density for Z is then given by
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14. Example Suppose X, Y are independent N(0,1). The density of is
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15. Moments The kth moment of a distribution is E(Xk). It is given by…
We are usually interested in 1st and 2nd moments (sometimes in 3rd and 4th)
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16. Moment Generating Functions
The moment generating function of a random variable X is
mX(t) exists if mX(t) < ∞ for |t| < t0 >0
If X is discrete
If X is continuous
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17. Examples X ~ Exponential(λ). The mgf of X is
X ~ Uniform(0,1). The mgf of X is
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18. Generating Moments from MGFs
Theorem
Let X be any random variable. If mX(t) < ∞ for |t| < t0 for some t0 > 0. Then
mX(0) = 1
etc. In general,
Where is the kth derivative of mX with respect to t.
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19. Examples
Suppose X ~ Exponential(λ). Find the mean and variance of X using its moment generating function.
Suppose X ~ N(0,1). Find the mean and variance of X using its moment generating function.
Suppose X ~ Binomial(n, p). Find the mean and variance of X using its moment generating function.
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20. Properties of Moment Generating Functions
mX(0) = 1.
If Y=a+bX, then the mgf of Y is given by
If X,Y independent and Z = X+Y then,
The last property can be generalized to a sum of n independent random variables.
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21. Uniqueness Theorem
If a moment generating function mX(t) exists for t in an open interval containing 0, it uniquely determines the probability distribution.
This can also be stated as follows:
Let X and Y are two random variables with mgf MX(t) and MY(t),
respectively. If MX(t) = MY(t) for all values of t, then X and Y have the same
probability distribution.
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22. Example
Find the mgf of X ~ N(μ,σ2) using the mgf of the standard normal random variable.
Suppose, , independent.
Find the distribution of X1+X2 using mgf approach.
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23. More on Normal Distribution
In general,
If X1, X2,…, Xn i.i.d N(0,1) then X1+ X2+…+ Xn ~ N(0,n).
If , ,…, then
If X1, X2,…, Xn i.i.d N(μ, σ2) then
Sn = X1+ X2+…+ Xn ~ N(nμ, nσ2) and
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24. Sum of Independent χ2(1) random variables
Recall: The Chi-Square density with 1 degree of freedom is the
Gamma(½ , ½) density.
If X1, X2 i.i.d with distribution χ2(1). Find the density of Y = X1+ X2.
In general, if X1, X2,…, Xn ~ χ2(1) independent then
X1+ X2+…+ Xn ~ χ2(n) = Gamma(n/2, ½).
Recall: The Chi-Square density with parameter n is
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25. Characteristic Function
The characteristic function, cX, of a random variable X is defined by:
The definition of the characteristics function is just like the definition of the
mgf, except for the introduction of the imaginary number
Using properties of complex numbers, we see that the characteristic
function can also be written as
The characteristics function unlike the mgf is always finite…
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26. Properties of Characteristic Function
The characteristics function has many nice properties similar to the mgf.
In particular, it can be used to generate moments.
Let X be any random variable with its first k moments finite. Then
cX(0) = 1
etc. In general,
Where is the kth derivative of cX with respect to s.
Let X and Y be independent random variables. Then
cX + Y (s) = cX(s) cY(s)
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27. Examples
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