The problem reads: Suppose P(X_1)=.75 and P(Y_2|X_1)=.40. What is the joint probability of X_1 and Y_2? This is how I answered it. P(X_1 and Y_2) =P(X_1)\times P(Y_1|X_1)=.75\times .40=0.3. What I don't understand is how do you get the P(Y_1|X_1)? I am totally new to Statistices and I need to understand each part of the process in order to get the whole concept. Can anyone help me to understand why the P and X exist and what they represent?

The problem reads: Suppose P(X_1)=.75 and P(Y_2|X_1)=.40. What is the joint probability of X_1 and Y_2? This is how I answered it. P(X_1 and Y_2) =P(X_1)\times P(Y_1|X_1)=.75\times .40=0.3. What I don't understand is how do you get the P(Y_1|X_1)? I am totally new to Statistices and I need to understand each part of the process in order to get the whole concept. Can anyone help me to understand why the P and X exist and what they represent?

Question
The problem reads: Suppose \(\displaystyle{P}{\left({X}_{{1}}\right)}={.75}\) and \(\displaystyle{P}{\left({Y}_{{2}}{\mid}{X}_{{1}}\right)}={.40}\). What is the joint probability of \(\displaystyle{X}_{{1}}\) and \(\displaystyle{Y}_{{2}}\)?
This is how I answered it. P(\(\displaystyle{X}_{{1}}\) and \(\displaystyle{Y}_{{2}}\)) \(\displaystyle={P}{\left({X}_{{1}}\right)}\times{P}{\left({Y}_{{1}}{\mid}{X}_{{1}}\right)}={.75}\times{.40}={0.3}.\)
What I don't understand is how do you get the \(\displaystyle{P}{\left({Y}_{{1}}{\mid}{X}_{{1}}\right)}\)? I am totally new to Statistices and I need to understand each part of the process in order to get the whole concept. Can anyone help me to understand why the P and X exist and what they represent?

Answers (1)

2021-01-11
\(\displaystyle{X}_{{1}}\) and \(\displaystyle{Y}_{{2}}\) both represent the occurrence of someevent, e.g. the number that comes up after rolling a fair die or if a person is a smoker or a non-smoker. \(\displaystyle{P}{\left({X}_{{1}}\right)}\) represents the probability that the event \(\displaystyle{X}_{{1}}\) occurs. \(\displaystyle{P}{\left(\frac{{Y}_{{2}}}{{X}_{{1}}}\right)}\) represents the probability that the event \(\displaystyle{Y}_{{2}}\) occurs given that the event \(\displaystyle{X}_{{1}}\) has occurred. In other words, we wantto know what chance \(\displaystyle{Y}_{{2}}\) will occur knowing that \(\displaystyle{X}_{{1}}\) has already happened.
The joint probability of \(\displaystyle{X}_{{1}}\) and \(\displaystyle{Y}_{{2}}\), denoted by \(\displaystyle{P}{\left({X}_{{1}}\cap{Y}_{{2}}\right)}\) or \(\displaystyle{P}{\left({Y}_{{2}}\cap{X}_{{1}}\right)}\)
\(\displaystyle{P}{\left({X}_{{1}}\cap{Y}_{{2}}\right)}={P}{\left({Y}_{{2}}\cap{X}_{{1}}\right)}\)
by the commutative property, representsthe probability that both events \(\displaystyle{X}_{{1}}\) and \(\displaystyle{Y}_{{2}}\) occur.
Use the following identity to compute
\(\displaystyle{P}{\left({Y}_{{2}}{\mid}{X}_{{1}}\right)}={\frac{{{P}{\left({X}_{{1}}\cap{Y}_{{2}}\right)}}}{{{P}{\left({X}_{{1}}\right)}}}}\)
The formula for \(\displaystyle{P}{\left({Y}_{{2}}{\mid}{X}_{{1}}\right)}\) can be derived in the following manner: If the event \(\displaystyle{X}_{{1}}\) occurs, then in order for the event \(\displaystyle{Y}_{{2}}\) to occur, then they both have to occur. This explains the numerator. Since we know that the event \(\displaystyle{X}_{{1}}\) has occurred and we are only interested in a small part of \(\displaystyle{X}_{{1}}\) (case where \(\displaystyle{Y}_{{2}}\) occurs when \(\displaystyle{X}_{{1}}\) has occured), then \(\displaystyle{X}_{{1}}\) represents the set of all possibleoutcomes. This explains the denominator.
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