Let x be the standard normal random variable. find the following:
a)P(0)<x<1.42)
b)P(-0.73<x<0)
c)P(-1.37<x<2.01)
d)P(x>1.13)
To find the probabilities for the given standard normal random variable, we can use the standard normal distribution table or a calculator with a built-in cumulative distribution function (CDF) for the standard normal distribution.
a) P(0 < x < 1.42)
To find this probability, we need to find the area under the standard normal curve between 0 and 1.42. We can use a standard normal distribution table or a calculator.
Using a standard normal distribution table:
The table provides the cumulative probability values for the standard normal distribution, usually denoted as "Z." To use the table, we need to convert the given values into Z-scores.
In this case, we can use the formula:
Z = (x - μ) / σ
Since x is given as 1.42 and it is a standard normal random variable, μ (mean) is 0 and σ (standard deviation) is 1.
Z = (1.42 - 0) / 1 = 1.42
From the standard normal distribution table, we can find the probability corresponding to a Z-score of 1.42. The value is 0.9222.
Therefore, P(0 < x < 1.42) = 0.9222.
b) P(-0.73 < x < 0)
Similarly, we need to find the area under the standard normal curve between -0.73 and 0. We can either use the standard normal distribution table or a calculator.
To use the table:
Convert the given values into Z-scores.
For x = -0.73:
Z = (-0.73 - 0) / 1 = -0.73
For x = 0:
Z = (0 - 0) / 1 = 0
Using the standard normal distribution table, find the cumulative probability values for both Z-scores.
P(Z < -0.73) = 0.2314
P(Z < 0) = 0.5000
Since we're interested in the probability between -0.73 and 0, we subtract the lower value from the higher value:
P(-0.73 < x < 0) = P(Z < 0) - P(Z < -0.73) = 0.5000 - 0.2314 = 0.2686.
Therefore, P(-0.73 < x < 0) = 0.2686.
c) P(-1.37 < x < 2.01)
Using the same steps as above, convert -1.37 and 2.01 into Z-scores:
For x = -1.37:
Z = (-1.37 - 0) / 1 = -1.37
For x = 2.01:
Z = (2.01 - 0) / 1 = 2.01
Find the cumulative probability values for both Z-scores:
P(Z < -1.37) = 0.0853
P(Z < 2.01) = 0.9772
Subtract the lower value from the higher value:
P(-1.37 < x < 2.01) = P(Z < 2.01) - P(Z < -1.37) = 0.9772 - 0.0853 = 0.8919.
Therefore, P(-1.37 < x < 2.01) = 0.8919.
d) P(x > 1.13)
To find this probability, we need to calculate the area under the standard normal curve to the right of 1.13.
Convert 1.13 into a Z-score:
Z = (1.13 - 0) / 1 = 1.13
Using the standard normal distribution table, find the cumulative probability value for a Z-score of 1.13:
P(Z > 1.13) = 1 - P(Z < 1.13)
From the table, P(Z < 1.13) = 0.8708. Therefore:
P(Z > 1.13) = 1 - 0.8708 = 0.1292.
Therefore, P(x > 1.13) = 0.1292.