let X~normal(0,16) and Y~normal(1,9), find P(X>Y)
To find P(X > Y), we need to compare the values of X and Y. However, because these variables are normally distributed, the comparison is not straightforward.
Here's how we can proceed step by step:
Step 1: Standardize the variables
To compare the two normal distributions, we need to standardize them by converting them into standard normal distributions (normal distribution with a mean of 0 and a standard deviation of 1). We can standardize a random variable Z using the formula Z = (X - μ) / σ, where X is the random variable, μ is the mean, and σ is the standard deviation.
For X ~ normal(0,16), the corresponding standardized variable Z1 is:
Z1 = (X - μ1) / σ1 = (X - 0) / 16 = X/16
For Y ~ normal(1,9), the corresponding standardized variable Z2 is:
Z2 = (Y - μ2) / σ2 = (Y - 1) / 9
Step 2: Convert the inequality
Since we want to find P(X > Y), we need to convert the inequality to the standardized variables. Therefore, we rewrite the inequality as Z1 > Z2.
Step 3: Find the probability
Now we can find the probability P(X > Y) by calculating the area under the standard normal curve for Z1 > Z2. This can be done using statistical software, a table of standard normal probabilities, or by using a calculator.
Alternatively, we can use the fact that the difference between two independent normal random variables, X and Y, is also normal with mean μ1 - μ2 and standard deviation sqrt(σ1^2 + σ2^2). In this case, the difference of X - Y follows a normal distribution with mean 0 - 1 = -1 and standard deviation sqrt(16 + 9) = sqrt(25) = 5. We can then find P(X > Y) as P(X - Y > 0) which means finding the probability of the standardized difference Z = (X - Y - (μ1 - μ2)) / sqrt(σ1^2 + σ2^2). In this case, we have Z = (X - Y - (-1)) / 5 = (X - Y + 1) / 5.
After standardizing, we have:
P(X > Y) = P(Z1 > Z2) = P((X/16) > ((Y - 1)/9)) = P(5*X - 5*Y + 5 > 0)
By using statistical software, a table of standard normal probabilities, or a calculator, you can find the probability P(5*X - 5*Y + 5 > 0) under the standard normal distribution.