# Cdf, expectation, and variance of a random continuous variable

• MHB
In summary: What is the probability that X is within one standard deviation from the mean?In summary, the probability that $X$ is within one standard deviation from the mean is .
Given the probability density function f(x) = b[1-(4x/10-6/10)^2] for 1.5 < x <4. and f(x) = 0 elsewhere.

1. What is the value of b such that f(x) becomes a valid density function

2. What is the cumulative distribution function F(x) of f(x)

3. What is the Expectation of X, E[X]

4. What is the Variance of X, Var[X]

5. What is the probability that X is within one standard deviation from the meanSo far, I've gotten b by integrating the function from 1.5 to 4 and setting it equal to 1, thus getting b = 3/5.
Plugging b into the function, I also integrated it from 1.5 to x, to get a cdf of (-4x^3/125 + 18x^2/125 + 48x/125 - 99/125)
To get the E[X], I integrated f(x) from 1.5 to 4 to get 2.4375
To solve for the variance i used the equation E[X^2] - E[X]^2 and got 0.3711

However, help would be appreciated for number 5 as I am not even sure where to start on that one.

Last edited:
Given the probability density function f(x) = b[1-(4x/10-6/10)^2] for 1.5 < x <4. and f(x) = 0 elsewhere.

1. What is the value of b such that f(x) becomes a valid density function

2. What is the cumulative distribution function F(x) of f(x)

3. What is the Expectation of X, E[X]

4. What is the Variance of X, Var[X]

5. What is the probability that X is within one standard deviation from the meanSo far, I've gotten b by integrating the function from 1.5 to 4 and setting it equal to 1, thus getting b = 3/5.
Plugging b into the function, I also integrated it from 1.5 to x, to get a cdf of (-4x^3/125 + 18x^2/125 + 48x/125 - 99/125)
To get the E[X], I integrated f(x) from 1.5 to 4 to get 2.4375
To solve for the variance i used the equation E[X^2] - E[X]^2 and got 0.3711

However, help would be appreciated for number 5 as I am not even sure where to start on that one.

The probability that $X$ is between two arbitrary values $a_1 \le a_2$ is:
$$P(a_1<X<a_2)=\int_{a_1}^{a_2}f(x)\,dx$$
The standard deviation $\sigma[X]$ is the square root of the variance:
$$\sigma[X]=\sqrt{\operatorname{Var}[X]}$$
So in this case:
$$P(\text{X is within one standard deviation from the mean}) \\= P(E[X]-\sigma[X]<X<E[X]+\sigma[X]) \\= \int_{E[X]-\sigma[X]}^{E[X]+\sigma[X]} f(x)\,dx$$

## 1. What is a CDF (Cumulative Distribution Function)?

A CDF is a function that maps the probability of a random continuous variable being less than or equal to a certain value. It is used to describe the overall behavior of a probability distribution and can be used to calculate the probability of a random variable falling within a specific range.

## 2. How is the expectation of a random continuous variable calculated?

The expectation, also known as the mean, of a random continuous variable is calculated by multiplying each possible value of the variable by its corresponding probability and then summing all of these products. This can also be represented mathematically as E(X) = ∫xf(x)dx, where x is the random variable and f(x) is its probability density function.

## 3. What does the variance of a random continuous variable represent?

The variance of a random continuous variable measures the spread or variability of the values of the variable around its mean. It is calculated by taking the average of the squared differences between each value and the mean. A higher variance indicates a wider range of values, while a lower variance indicates a more concentrated distribution around the mean.

## 4. How is the variance of a random continuous variable calculated?

The variance of a random continuous variable can be calculated using the formula Var(X) = E[(X - μ)^2], where μ is the mean of the variable. This formula can also be written as Var(X) = ∫(x - μ)^2f(x)dx, where x is the random variable and f(x) is its probability density function.

## 5. Can the CDF, expectation, and variance of a random continuous variable be used to make predictions?

Yes, the CDF, expectation, and variance of a random continuous variable can be used to make predictions about the behavior of the variable. The CDF can be used to calculate the probability of a certain outcome, while the expectation and variance can provide information about the average and variability of the values. These measures can be used to make informed decisions and predictions in various fields such as finance, economics, and science.

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