Search
STATLECT > Probability distributions

Probability distributions - Multiple choice test 2

Take a multiple choice test on the fundamentals of probability theory.

When you give a right answer, it is marked with nav_button.

When you give a wrong answer, it is marked with nav_button and the right answer is marked with nav_button.

When you answer a question, your score is updated and you can read an explanation of the right answer before going to the next question.

nav_button Start taking the test now.

Score: 0% (0 out of 0)

Question 1. Let X be an absolutely continuous random variable, with support [eq1] and probability density function [eq2]. If [eq3] for [eq4], then X has a:

nav_button normal distribution

nav_button binomial distribution

nav_button Bernoulli distribution

nav_button uniform distribution

nav_button Explanation | nav_button Next question

Explanation. A uniform random variable is characterized by the fact that its probability density function is constant on its support. For more details, see the lecture entitled uniform distribution.

nav_button Next question

Question 2. Let X be a uniform random variable, with support [eq5] and probability density function [eq6]. When $xin R_{X}$:

nav_button [eq7]

nav_button [eq8]

nav_button [eq9]

nav_button [eq10]

nav_button Explanation | nav_button Next question

Explanation. If X has a uniform distribution, its probability density function must be constant on its support. Furthermore, the integral of the probability density function over the support must be equal to 1:[eq11]For more details, see the lectures entitled uniform distribution and legitimate probability density functions.

nav_button Next question

Question 3. Consider a uniform random variable X, with probability density function:[eq12]The expected value of $X^{2}$ is:

nav_button [eq13]

nav_button [eq14]

nav_button [eq15]

nav_button [eq16]

nav_button Explanation | nav_button Next question

Explanation. The expected value of $X^{2}$ can be calculated as follows:[eq17]For more details, see the lecture entitled uniform distribution.

nav_button Next question

Question 4. A random variable X is uniformly distributed between $2$ and $5$. Its probability density function is:[eq18]where $c$ is a constant. [eq19] is a legitimate probability density function if:

nav_button $c= rac{1}{3}$

nav_button $c= rac{1}{5}$

nav_button $c=3$

nav_button $c=1$

nav_button Explanation | nav_button Next question

Explanation. [eq20] is a legitimate probability density function if the integral of [eq21] over its support is equal to 1. The integral is:[eq22]So, it must be:[eq23]which implies:[eq24]For more details, see the lecture entitled legitimate probability density functions.

nav_button Next question

Question 5. Let X be an absolutely continuous random variable with support $R_{X}=U{211d} $. X has a standard normal distribution if its probability density function [eq25] is:

nav_button [eq26]

nav_button [eq27]

nav_button [eq28]

nav_button [eq29]

nav_button Explanation | nav_button Next question

Explanation. A standard normal random variable is a normal random variable with zero mean and unit variance. Its probability density function is:[eq30]See the lecture entitled normal distribution for further details.

nav_button Next question

Question 6. Let X be an absolutely continuous random variable having a standard normal distribution. Let:[eq31] The probability density function of Y is:

nav_button [eq32]

nav_button [eq33]

nav_button [eq34]

nav_button [eq35]

nav_button Explanation | nav_button Next question

Explanation. X is a normal random variable with zero mean and unit variance. Y, being a linear function of X, is a normal random variable with mean $mu =3$ and variance [eq36]. Therefore, its probability density function is: [eq37]See the lecture entitled normal distribution for further details.

nav_button Next question

Question 7. The moment generating function of a standard normal random variable X is:

nav_button [eq38]

nav_button [eq39]

nav_button [eq40]

nav_button [eq41]

nav_button Explanation | nav_button Next question

Explanation. The moment generating function of a standard normal random variable is:[eq42]See the lecture entitled normal distribution for further details.

nav_button Next question

Question 8. The moment generating function of a standard normal random variable X is:[eq43]Therefore, the moment generating function of a normal random variable Y with mean mu and variance sigma^2 is:

nav_button [eq44]

nav_button [eq45]

nav_button [eq46]

nav_button [eq47]

nav_button Explanation | nav_button Next question

Explanation. Y is equal to a linear function of a standard normal random variable X:[eq48]Therefore, its moment generating function is:[eq49]See the lecture entitled normal distribution for further details.

nav_button Next question

Question 9. Let X be a chi-square random variable with n degrees of freedom ($nin U{2115} $). The support of X is:

nav_button [eq50]

nav_button $R_{X}=U{211d} $

nav_button [eq51]

nav_button [eq52]

nav_button Explanation | nav_button Next question

Explanation. The support of a chi-square random variable (i.e. the set of values it can take) is the set of positive real numbers:[eq53]See the lecture entitled chi-square distribution for further details.

nav_button Next question

Question 10. If a function [eq54] is proportional to another function [eq55] we write:[eq56]which means that:[eq57]where $c$ is a constant that does not depend on x. Now, let X be a chi-square random variable with n degrees of freedom and [eq58] its probability density function. When [eq59]:

nav_button [eq60]

nav_button [eq61]

nav_button [eq62]

nav_button [eq63]

nav_button Explanation | nav_button Next question

Explanation. When [eq64], the probability density function of a chi-square random variable with n degrees of freedom is:[eq65]But[eq66]is a constant. Thus:[eq67]See the lecture entitled chi-square distribution for more details.

nav_button Next question

Question 11. Let X be a chi-square random variable with n degrees of freedom. Its expected value is:

nav_button [eq68]

nav_button [eq69]

nav_button [eq70]

nav_button [eq71]

nav_button Explanation | nav_button Next question

Explanation. The expected value of a chi-square random variable with $n $ degrees of freedom is:[eq72]See the lecture entitled chi-square distribution for more details.

nav_button Next question

Question 12. Let X be a chi-square random variable with n degrees of freedom. Its variance is:

nav_button [eq73]

nav_button [eq74]

nav_button [eq75]

nav_button [eq76]

nav_button Explanation | nav_button Next question

Explanation. The variance of a chi-square random variable with n degrees of freedom is:[eq77]See the lecture entitled chi-square distribution for more details.

nav_button Next question

Question 13. Let X_1 be a chi-square random variable with $2$ degrees of freedom and X_2 be a chi-square random variable with $3$ degrees of freedom. Also, let X_1 and X_2 be mutually independent. Then:

nav_button the difference $X_{2}-X_{1}$ is a Gamma random variable with 1 degree of freedom

nav_button the product $X_{1}X_{2}$ is a chi-square random variable with $6$ degrees of freedom

nav_button the ratio $X_{1}/X_{2}$ is a normal random variable

nav_button the sum $X_{1}+X_{2}$ is a chi-square random variable with $5$ degrees of freedom

nav_button Explanation | nav_button Next question

Explanation. Let X_1 be a chi-square random variable with $n_{1}$ degrees of freedom and X_2 another chi-square random variable with $n_{2} $ degrees of freedom. If X_1 and X_2 are independent, then their sum has a chi-square distribution with $n_{1}+n_{2}$ degrees of freedom. See the lecture entitled chi-square distribution for more details.

nav_button Next question

Question 14. Let X be a chi-square random variable with 1 degree of freedom. Its moment generating function is:[eq78]and it is defined for $t<1/2$. If Y is another chi-square random variable with $3$ degrees of freedom, then its moment generating function must be:

nav_button [eq79]

nav_button [eq80]

nav_button [eq81]

nav_button [eq82]

nav_button Explanation | nav_button Next question

Explanation. A chi-square random variable with $3$ degrees of freedom can be viewed as a sum of $3$ independent chi-square random variables with $1 $ degree of freedom. The moment generating function of a sum of independent random variables is the product of their moment generating functions. Thus:[eq83]See the lectures entitled chi-square distribution and moment generating function of a random variable for more details.

nav_button Next question

Question 15. Let X be a chi-square random variable with $2$ degrees of freedom and let Y be a standard normal random variable. Let X and Y be mutually independent. Define:[eq84]Then Z has a:

nav_button Student's t distribution

nav_button chi-square distribution with $2$ degrees of freedom

nav_button non-standard distribution

nav_button chi-square distribution with $3$ degrees of freedom

nav_button Explanation

Explanation. The square of a standard normal random variable has a chi-square distribution with 1 degree of freedom. Therefore, $Y^{2}$ has a chi-square distribution with 1 degree of freedom and Z is the sum of two independent chi-square random variables. Thus, $Z,$ has also a chi-square distribution. The degrees of freedom of Z are equal to the sum of the degrees of freedom of X and $Y^{2}$. See the lecture entitled chi-square distribution for more details.

by
About | Contacts | Privacy and terms of use | Sitemap