Please answer these 3 questions 1) The difference between Probability mass funct

Question

Please answer these 3 questions

1)

The difference between Probability mass function (PMF) and cumulative distribution function (CDF)?

I mean in joint probability: P(x,y) = P(X=x, Y=x)

and F(x,y) = P(X=x, Y=y)

so shouldnt F(x,y) = P(x,y)?

2)

So I just made this table of probabilities:

Is the 1/2, 1/4, 1/8, 1/8 probabilities basically: F(x,y), f(x,y), or P(x,y)??

Is the X=0 row....which is equal to 5/8....be the density functions Fx(0), fX(0), or PX(0)?

3)

difference between p(x,y) and P(x,y)...... are they the same?

Explanation / Answer

1) In terms of Joint PMF of Random variable X and Y

p(x,y) =P(X =x, Y =y)

and the joint CDF is

F(a,b) =P{X<= a, Y<= b}

so you should not write  F(x,y) = P(x,y)

Q2) P(X=0, Y = 0) = 1/2

P(X=1, Y =0) = 1/4

P(X=0, Y=1) = 1/8

P(X=1, Y=1) = 1/8

all these are conditional probabilities

When X =0, the value 5/8 shows the marginal probability of X when X =0 denoted as px(xi) = px(0)

3)p(x,y) shows joint probability mass function and P(x,y) shows the conditional probability

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