Continuous random variables and distributions

How do you know if a random variable X is continuous?

It’s set of possible values either consists of all numbers in a single interval or all numbers in. a union of disjoint intervals

P\[X = c\] = 0, for any possible value c of X

When is a a function f, a probability density function of a random variable?

Iff f(x) ≥ 0 for all x and

∫(infinity or negative infinity) f(x) dx = 1

(The area under the curve of f is one)

What is the expression for the area under the curve of f between a and b?

P\[ a ≤ X ≤ b\]

What is the probability density function of a continuous RV, X?

It is a function f that satisfies for any two numbers a≤b

∫(b to a) f(x) dx = P\[a≤X≤b\]

What is the CDF of a continuous random variable X with PDF ƒ?

F(x) = P\[X ≤ x\] = ∫(x to -infinity) f(t) dt

What conditions should a cumulative distribution function F satisfy?

If X be a continuous random variable with CDF F and PDF f, what does F’(x) equal?

At every x where F’(x) exists then;

F’(x) = f(x)

If X be a continuous random variable with CDF F and PDF f, what does P\[X > a\] equal?

1 - P\[X ≤ a\] = 1 - F(a)

If X be a continuous random variable with CDF F and PDF f, what does P\[a ≤ X ≤ B\] equal?

P\[X≤b\] - P\[X≤a\] = F(b) - F(a)

Let X be a continuous random variable with density function f(x), what is the expectation?

E\[X\] = µ = ∫(infinity to -infinity) (x - µ)^2 f(x) dx

Let X be a continuous random variable with density function f(x), then what is the variance of X?

V\[X\] = E\[(X-µ)^2\] = ∫(infinity to -infinity) (x-µ)^2 f(x) dx

What is the standard deviation of X if X is a continuous random variable with density function f(x)?

σ = √(V\[X\])

When does the E\[X\] exist for X being a continuous random variable with density function f(x)?

∫(infinity to -infinity) |x| f(x) dx < infinity

If h(x) is a function of X, where X is a random continuous variable with density function f(x), what does E\[h(X)\] equal?

∫(infinity to -infinity) h(x) f(x) dx

What is the shortcut formula for the variance of X if X is a continuous RV?

V\[X\] = E\[X^2\] - E\[X\]^2

If X and Y are two continuous RVs, a and b are two fixed numbers, what is E\[aX + b\] equal to?

E\[aX + b\] = aE\[X\] + b

If X and Y are two continuous RVs, a and b are two fixed numbers, what is E\[aX + bY\]?

E\[aX + bY\] = aE\[X\] + bE\[Y\]

If X and Y are two continuous RVs, a and b are two fixed numbers, what is V\[aX + b\]?

a^2 V\[X\]