Continuous Probability - Advanced Engineering Math - Tutorial Slides, Slides of Engineering Mathematics

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Continuous Probability

• Continuous Random Variables and Distributions:

A random variable X and its distribution are continuous if its distribution function can be given by an integral:

Distribution function

Then, the probability function (density of the distribution)

for every x at which f(x) is continuous.

• the probability corresponding to an interval:
• Properties:

• The mean or expected value of a continuous random variable:

• E(X)=

• Mean (Expectation) of a Distribution

• E[c X] = c E[X]
• E[X+Y] = E[X]+E[Y]
• In the second property, X and Y are not necessarily independent
• Var(c X) = c^2 Var(X)
• Var(X+Y) = Var(X) + Var(Y)
• In the second property, X and Y must be independent
• Properties of Mean and Variance (same with discrete random variables)

Question 1 ( example 5 in book) :

• Method 1:
• X～f(x), - ∞< x< +∞, Y=g(X) is the function of random variable X, then
• and the density of Y is
• Functions of random variables

• Random variable X is uniformly distributed on [-1, 1], 𝑌𝑌 = 𝑋𝑋 2 , then F(y) and f(y)?
• Solution:

Question 2:

• Random variable X is uniformly distributed on [0, 1], 𝑌𝑌 = 𝑋𝑋 2 , then F(y) and f(y)?
• Solution: g(X)=x^2 is monotonous and derivable on [0, 1], and

Question 3: