Let Y=X2. Calculate the following probabilities:
P(Y=0)=
P(Y=1)=
P(Y=2)=
To calculate the probabilities, we need to know the distribution of the random variable X. However, if we assume X is continuous and uniformly distributed between -∞ and ∞, we can calculate the probabilities as follows:
P(Y = 0) = P(X² = 0)
Since X² = 0 only when X = 0, we can say:
P(Y = 0) = P(X = 0)
Assuming X has a continuous uniform distribution, the probability of X taking any specific value is 0. Therefore:
P(X = 0) = 0
P(Y = 1) = P(X² = 1)
Since X² = 1 only when X = 1 or X = -1, we can say:
P(Y = 1) = P(X = 1) + P(X = -1)
Again, assuming X has a continuous uniform distribution, the probability of X taking any specific value is 0. Therefore:
P(X = 1) = 0
P(X = -1) = 0
Thus:
P(Y = 1) = P(X = 1) + P(X = -1) = 0 + 0 = 0
Similarly,
P(Y = 2) = P(X² = 2)
Since X² = 2 has no real solutions, we can conclude that:
P(Y = 2) = 0
To calculate the probabilities P(Y=0), P(Y=1), and P(Y=2), we need to consider the probability distribution of the random variable Y, which is defined as Y = X^2, where X is another random variable.
To calculate these probabilities, we need to know the probability distribution of X. Without this information, we cannot directly calculate the probabilities P(Y=0), P(Y=1), and P(Y=2).
The probability distribution of X is needed to determine the likelihood of different values of X occurring, which in turn determines the probabilities of Y taking on specific values. Without the probability distribution of X, we cannot proceed with the calculations.
For example, if X is a continuous random variable, we need its probability density function (PDF). If X is a discrete random variable, we need its probability mass function (PMF).
Once we have the probability distribution of X, we can use it to calculate the probabilities P(Y=0), P(Y=1), and P(Y=2) by substituting the values into the formula for Y and then evaluating the probabilities based on the distribution of X.
In summary, to calculate the probabilities P(Y=0), P(Y=1), and P(Y=2), we need to have the probability distribution of X. Without this information, we cannot provide direct calculations for these probabilities.