How did the question on the probability density function go

Urgently please do any of you know how to do this question?

In the first question, by definition, the probability density function integrates over the whole plane as 1, and based on this you can find the unknown c. The sufficient condition for the independence of the two variables is fX(x)*fY(y)=f(x,y). Once the two are found, multiplying them together and comparing them to f(x,y) leads to the conclusion.

The second question, according to the formula of the correlation coefficient to find each unknown quantity: E(XY), E(X), E(X²), E(Y), E(Y²), and then directly substitute into the calculation can be. Note: The expected square is best written in the form [E(X)]². This way of writing E²(X) is not the standard way for me to write it, and it may be misjudged.

Probability density function, to find the process, especially the derivative of that step

This topic does not require complex calculations. First of all, the linear operation of mutually independent normal distribution is still normal distribution, the linear function of normal distribution (aX+b) is also normal distribution. Then Z=X-2Y+1 must obey the normal distribution, and EZ=EX-2EY+1=2-2×2+1=-1, DZ=DX+4DY=0.2+4×0.2=1, so Z~N(-1,1), so that you can directly write its probability density.

What are the steps to find the probability density function?

The distribution function is transformed into a probability density, and the probability density can be found by simply taking the derivative of the distribution function.

If the probability density is the probability density of the continuous type, then find the distribution function directly on the probability density direct integration can get the corresponding distribution function.

If the probability density is a segmented function, then we have to start from the definition of the distribution function to find the distribution function.

Note that the distribution function is cumulative. A segment-by-segment accumulation of the probabilities gives us the distribution inclusive tax.

So the probability density for this question:

When x<0 F(x)=∫(–∞,x)f(x)dx=0,

When 0<=x<1, F(x)=∫(o,x)tdt=(x^2)/2

When 1<=x<2,F(x)=∫(o,1) tdt+∫(1,x)2-tdt=2x-(x^2)/2-1.

When x>=2 F(x)=1.


Properties of the distribution function:

How do you do this question on probability density functions?

\int_{-\infty}^{+\infty}f(x)dx=1 Calculate k=-1/2

F(x)=\int_{-\infty}^xf(t)dt in three intervals x<0,0<=x<=2,x>2, and each of the other three.