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```Homework Set #5
Problem 1
X has probability density function as shown below.
fX ( x )
2
⎧2 x , 0 ≤ x ≤ 1
f X (x) = ⎨
otherwise
⎩0,
0
1
X
Calculate the mean value m X , variance σ 2X and second initial moment E[X 2 ] . Verify
the relaton E[X 2 ] = m 2X + σ 2X .
Problem 2
X has uniform distribution between 2 and 3. Consider a new variable Y = X 3 .
Sketch the function Y(X) .
Find the probability density function of Y.
Calculate the mean value and variance of X.
Using the probability density function found in (b), calculate the mean value and
variance of Y.
(e) Verify that m Y and σ 2Y can be obtained also as
3
3
3
2
m Y = x f X (x) dx and σ Y = (x 3 − m Y ) 2 f X (x) dx .
(a)
(b)
(c)
(d)
∫
2
∫
2
Problem 3
Consider two discrete random variables X1 and X 2 , with the joint probability mass
function shown in the figure below. (Notice that the distribution is concentrated at four
points, with equal probability 0.25 at each point).
X2
0.25
1
0.25
-1
0.25
0
1
X1
0.25
-1
(a) Are X1 and X 2 independent? Briefly explain why or why not.
(b) Find the mean values m1 and m 2 , the variances σ12 and σ 22 , and the correlation
coefficient ρ between X1 and X 2 .
```