Let's solve this problem graphically. Here we have the following equation:
So we can rewrite this as:
So the solution to the equation is the x-value at which the functions f and g intersect. In other words:
Using graphing calculator, we get that this value occurs at:
The solutions to the questions are
The probability density function is given as:
f(x)= xe^ -x for x>0
The probability is represented as:
So, we have:
Using an integral calculator, we have:
Expand the expression
Evaluate the expressions
P(2 < x < 4) =-0.092 +0.406
Evaluate the sum
P(2 < x < 4) = 0.314
Hence, the probability that X is between 2 and 4 is 0.314
<h3>Find the probability that the value of X exceeds 3</h3>This is represented as:
Using an integral calculator, we have:
Expand the expression
Evaluate the expressions
P(x > 3) =0 + 0.199
Evaluate the sum
P(x > 3) = 0.199
Hence, the probability that X exceeds 3 is 0.199
<h3>Find the expected value of X</h3>This is calculated as:
So, we have:
This gives
Using an integral calculator, we have:
Expand the expression
Evaluate the expressions
E(x) = 0 + 2
Evaluate
E(x) = 2
Hence, the expected value of X is 2
<h3>Find the Variance of X</h3>This is calculated as:
V(x) = E(x^2) - (E(x))^2
Where:
This gives
Using an integral calculator, we have:
Expand the expression
Evaluate the expressions
E(x^2) = -0 + 6
This gives
E(x^2) = 6
Recall that:
V(x) = E(x^2) - (E(x))^2
So, we have:
V(x) = 6 - 2^2
Evaluate
V(x) = 2
Hence, the variance of X is 2
Read more about probability density function at:
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<u>Complete question</u>
A random variable X with a probability density function f(x)= xe^ -x for x>0\\ 0& else
a. Find the probability that X is between 2 and 4
b. Find the probability that the value of X exceeds 3
c. Find the expected value of X
d. Find the Variance of X