You are given that a continuous random variable X is Normal/Gaussian with µ= 2 and s^2= 4. a) Write a mathematical expression for the probability density function (pdf) of X.Write the cumulative density function (cdf). Include actual numbers. b) What is the probability that X is greater than 4 [P{X > 4}]? What is the probabilitythat X is less than 1 [P{X < 1}]? Show your work. Hint: Drawing a rough sketch of theGaussian pdf may be helpful in checking your answers. c) Determine P{1 ≤ X ≤ 4}. Show your work.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
You are given that a continuous random variable X is Normal/Gaussian with µ= 2 and
s^2= 4.
a) Write a mathematical expression for the
Write the cumulative density function (cdf). Include actual numbers.
b) What is the probability that X is greater than 4 [P{X > 4}]? What is the probability
that X is less than 1 [P{X < 1}]? Show your work. Hint: Drawing a rough sketch of the
Gaussian pdf may be helpful in checking your answers.
c) Determine P{1 ≤ X ≤ 4}. Show your work.
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