Question

Two independent random variates X and Y are both normally distributed
with means 1 and 2 and standard deviations 3 and 4
respectively. If
Z=X - Y, write the probability density function of Z. Also state the median, s.d. and mean of
the distribution of Z. Find Probability (Z +1 <0).​

Answer 1

Answer:

This was a not a proof but an insight on how independent random variables affect variances on subtraction. Now calculate variance of $Z^{\prime}$ where $Z^{\prime}=X+Y$.

You'll get the answer $4.166$

Step-by-step explanation:

Step 1: A probability distribution that is symmetric around the mean is the normal distribution, sometimes referred to as the Gaussian distribution. It demonstrates that data that are close to the mean occur more frequently than data that are far from the mean.

Step 2: If  X be a random variable for a six sided die

$\begin{aligned}& X=\{1,2,3,4,5,6\} \\& E\left[X^2\right]=\frac{1}{6}\left(1^2+2^2+3^2+4^2+5^2+6^2\right)=15.166 \\& E[X]=\frac{1}{6}(1+2+3+4+5+6)=3.5 \\& (E[X])^2=3.5^2=12.25 \\& \sigma_X^2=E\left[X^2\right]-(E[X])^2=15.166-12.25 \\& \sigma_X^2=2.916\end{aligned}$

If Y be a random variable of a four sided die

$\begin{aligned}& Y=\{1,2,3,4\} \\& \sigma_Y^2=1.25\end{aligned}$

We only have information on X and Y in terms of the factors that produce values inside a certain domain.

Now we introduce a dependent random variable Z=X-Y

Since we are concurrently tossing both dice, both are independent random variables, and the outcome of one die has no bearing on the outcome of the other.

Z will depend on values of X and Y and will be equal to X-Y.

$\begin{aligned}& Z=\{-3,-2,-1,0,1,2,3,4,5\} \\& P(-3)=P(5)=\frac{1}{24} \\& P(-2)=P(4)=\frac{2}{24} \\& P(-1)=P(3)=\frac{3}{24} \\& P(2)=P(1)=P(0)=\frac{4}{24} \\& E[Z]=1 \\& E\left[Z^2\right]= \\& \frac{1}{24}\left((-3)^2+5^2\right)+\frac{2}{24}\left((-2)^2+4^2\right)+\frac{3}{24}\left((-1)^2+3^2\right)+\frac{4}{24}\left(0^2+1^2+\right. \\& =\frac{34}{24}+\frac{40}{24}+\frac{30}{24}+\frac{20}{24}=\frac{124}{24}=\frac{31}{6}=5.166 \\& \sigma_Z^2=5.166-1=4.166\end{aligned}$

Let's now apply the identification to see whether we are receiving the same response.

$\begin{aligned} \sigma_Z^2 & =\sigma_X^2+\sigma_Y^2 \\ \sigma_Z^2 & =2.916+1.25=4.166\end{aligned}$

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