How can I solve this:

Random variables $X,Y$ ~ Unif$(0, 1)$ are independent. Calculate the probability density function of sum $X + 3Y$.

I couldn't find a sum for uniformally distributed random variables. I assume I have to go straight to the PDF and solve it that way.

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  • 3
    Look up pdf convolution. – J.G. Apr 3 at 6:09
  • An alternative is via the cumulative distribution function in several intervals such as $(-\infty,0], (0,1], (1,3], (3,4], (4,\infty)$ and then take the derivative – Henry Apr 3 at 7:11

One way to avoid explicit convolution (although convolution is always involved, in the end), is to define $Z = 3Y \sim \text{Uniform}(0, 3)$, and look at the distribution of $(X, Z)$ in the $x$-$z$ plane: a uniformly distributed rectangle.

Within this rectangle, the bands of equal values of $X+Z$ correspond to diagonal stripes. The lengths of these stripes, where $X+Z =$ some value $w$, is proportional to the value of the PDF $f_{X+Z}(w)$. All you need to do, then, is to find the proportionality constant that makes it a PDF; that is, it must integrate to $1$.

Easy Understanding of Convolution The best way to understand convolution is given in the article in the link,using that

I am going to solve the above problem and hence you could follow the same for any similar problem such as this with not too much confusion.

$Z = X+ 3Y$ where X and Y are U(0,1).

I am going to define a new variable W where W is distributed according to U(0,3)

Thus $Z = X + 3Y = X+ W$ where X is U(0,1) and W is U(0,3).

Now I am going define the bounds

$t_{X_0} = 0$

$t_{X_1} = 1$

$t_{W_0} = 0$

$t_{W_1} = 3$

Thus $$f_Z(z) = 0, z \le t_{X_0}+t_{W_0} ,$$

$$f_Z(z) = \int_{max(t_{W_0}, t-t_{X_1})}^{min(t_{W_1}, t-t_{X_0})} f_W(w)f_X(z-w)dw, \text{ } t_{X_0}+t_{W_0} \le z \le t_{X_1}+t_{W_1},$$

$$f_Z(z) = 0, z \ge t_{X_1}+t_{W_1} ,$$

These translate to the following:

$$f_Z(z) = 0, z \le 0 ,$$

$$f_Z(z) = \int_{max(0, z-1)}^{min(3, z)} f_W(w)f_X(z-w)dw, \text{ } 0\le z \le 4,$$

$$f_Z(z) = 0, z \ge 4 ,$$

$f_W(w) = \frac{1}{3}$ as $W$ is $U(0,3)$.

$f_X(x) = 1 $ as $X$ is $U(0,1)$,

The middle one needs to be split into three intervals, and they are a) $0\le z\le 1$, b) $1\le z\le 3$, and c) $3\le z\le 4$.

Thus

$f_Z(z) = \int_{0}^{z}\frac{1}{3}dw = \frac{z}{3}$, $0\le z\le 1$

$f_Z(z) = \int_{z-1}^{z}\frac{1}{3}dw = \frac{1}{3}$, $1\le z\le 3$

$f_Z(z) = \int_{z-1}^{3}\frac{1}{3}dw = \frac{4-z}{3}$, $3\le z\le 4$

Sanity check is to find if $\int_{0}^{4} f_Z(z) = 1$ which it is in this case and hence the solution.

Goodluck

So we define $Z=X+3Y$ and the distribution is therefore

$P(Z) dZ = \int_0^1 dX \int_0^1 dY \delta(X+3Y-Z)$

where $\delta(X+3Y-Z)$ is the delta function that enforces the constraint $Z=X+3Y$. The double integral is effectively summing over all possible combinations of $X$ and $Y$ that produce a value of $X+3Y$ that equals $Z$.

We may be tempted to remove the delta function by simply setting $Y=(Z-X)/3$. But this is not enough to solve the problem as the finite ranges of $X,Y$ can limit the number of ways they can combine to produce a given $Z$. Note that the support of $Z$ is between $0$ and $4$.

As a result the distribution of $Z$ can depend on the value of $Z$, which may seem counterintuitive given that $X$ and $Y$ both have distributions that are uniform with a probability density that is independent of $X$ and $Y$. But it is not counterintuitive once you realise that the probability of $Z=4$ which requires $X=1$ and $Y=1$ must be less than the probability of $Z=2$ which can be created by an infinite number of combinations of values of $X$ and $Y$.

We need to consider each range of values separately:

If $1\le Z \le 3$ then $Y$ must go from $(Z-1)/3$ to $Z/3$. The length of the $Y$ range is independent of $Z$ and equals $1/3$. So in this range $P(Z) = 1/3$.

If $Z<1$ the $Y$ range is $[0,Z/3]$. So the length of the $Y$ range is $Z/3$. So $P(Z)=Z/3$.

If $Z>3$ the $Y$ range is $[(Z-1)/3,1]$ and so the length of the range is $(4-Z)/3$. So the probability density is $P(Z)=(4-Z)/3$.

So we have five ranges:

If $Z<0$, $P(Z)=0$

If $0 \le Z < 1$, $P(Z)=Z/3$

If $1 \le Z < 3$, $P(Z)=1/3$

If $3 \le Z < 4$, $P(Z)=(4-Z)/3$

If $Z \ge 4$, $P(Z)=0$

I have provided Python code below that simulates the random variable $Z$ and plots its distribution.

enter image description here

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