Consider an RV X with the following density given parameters α and γ .
f (x) = {(1/γ)e^(-(x-α)/γ) if x>α
{0 if x ≤ α.
a. Find E[X].
b. Find V[X].
c. If α = 4, γ = 2, find P (X > 8).
d. Find P (X < α + 3γ ).
e. Find the cdf for X , F (x).
To find the answer to each of these questions, we'll need to use the given density function for the random variable X and apply the appropriate formulas.
a. Find E[X]:
The expected value (E[X]) of a random variable can be found by integrating the product of the variable's values and its probability density function (pdf).
E[X] = ∫ x * f(x) dx (from negative infinity to positive infinity)
In this case, we have a piecewise function, so we need to consider the ranges separately:
E[X] = ∫ (x * (1/γ)e^(-(x-α)/γ)) dx (from α to positive infinity)
Since the density is zero for x ≤ α, we only integrate over the range where x > α.
b. Find V[X]:
The variance (V[X]) of a random variable is a measure of its dispersion and is calculated as the expectation of the squared difference between each value and the expected value.
V[X] = E[(X - E[X])^2]
c. If α = 4, γ = 2, find P(X > 8):
To find the probability that X is greater than a particular value, we integrate the density function over the range of values that satisfy the condition.
P(X > 8) = ∫ f(x) dx (from 8 to positive infinity)
In this case, we use the density function for x > α, since the probability is zero for values less than or equal to α.
d. Find P(X < α + 3γ):
To find the probability that X is less than a particular value, we integrate the density function over the range of values that satisfy the condition.
P(X < α + 3γ) = ∫ f(x) dx (from negative infinity to α + 3γ)
e. Find the cumulative distribution function (cdf) for X, F(x):
The cumulative distribution function gives the probability that the random variable X takes a value less than or equal to a particular value x. It is the integral of the pdf up to x.
F(x) = ∫ f(t) dt (from negative infinity to x)
In this case, we'll need to consider the two ranges of the density function separately, integrating over x > α and x ≤ α.