Probability of Normal Random VariablesLet’s say that you need to find the cumulative probability for a random variable X which is normally distributed. You do not know what the value of X is or, for that matter, what the value of µ and σ is. You only know that X = µ + σ. Can you find the cumulative probability, i.e. the probability of the variable being less than µ + σ?
Question
Probability of Normal Random VariablesLet’s say that you need to find the cumulative probability for a random variable X which is normally distributed. You do not know what the value of X is or, for that matter, what the value of µ and σ is. You only know that X = µ + σ. Can you find the cumulative probability, i.e. the probability of the variable being less than µ + σ?
Solution
Yes, you can find the cumulative probability of the variable being less than µ + σ even if you don't know the specific values of µ (mean) and σ (standard deviation). This is because of the properties of the standard normal distribution.
In a standard normal distribution, the mean (µ) is 0 and the standard deviation (σ) is 1. So, if X = µ + σ, then X is 1 standard deviation above the mean.
The cumulative probability of a variable being less than 1 standard deviation above the mean in a standard normal distribution is approximately 0.8413 or 84.13%. This is a known value from the standard normal distribution table.
So, without knowing the specific values of µ and σ, we can say that the cumulative probability of the variable being less than µ + σ is approximately 0.8413 or 84.13%.
Similar Questions
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