Question Type 1: Finding the probability of certain outcomes of random variables Exercises

For $X$ with pmf $P(X = x) = 0.5^x(1 - 0.5)$ for $x = 0, 1, 2, \dots$, calculate $P(X \le 1)$.

A random variable $X$ has probability mass function $P(X=0)=0.3$, $P(X=1)=0.4$, $P(X=2)=0.3$. Find $E(X)$.

Let $X$ be a discrete random variable with probability mass function $P(X = x) = \frac{6}{11x}$ for $x = 1, 2, 3$.

Find $P(X > 1)$.

Let $X$ be a discrete random variable with probability mass function $P(X=x)=kx^2$ for $x=1,2,3,4$. Determine the constant $k$.

A loaded six-sided die has probabilities $P(X = i) = \frac{i}{21}$ for $i = 1, 2, 3, 4, 5, 6$. Find $E(X)$.

A random variable $X$ has a probability mass function $P(X = x) = \frac{x + 1}{15}$ for $x = 0, 1, 2, 3, 4$.

Calculate $\text{E}(X)$.

A random variable $X$ has the probability distribution $P(X = 0) = 0.1$, $P(X = 1) = 0.2$, $P(X = 2) = 0.3$, and $P(X = 3) = 0.4$.

Find $P(X \le 2)$.

A spinner has four sectors numbered 1, 2, 3 and 4. The probability of landing on each sector is $P(1) = p, P(2) = 0.1, P(3) = 0.3$ and $P(4) = q$.

The cost to play the game is $2 and the player wins the number of dollars shown on the sector.

Find the values of $p$ and $q$ such that the game is fair.

Let $X$ be a discrete random variable taking values $-1, 2, 5$ with probabilities $P(X = -1) = p$, $P(X = 2) = q$, and $P(X = 5) = 1 - p - q$.

Determine the possible values of $p$ and $q$ such that the game is fair (i.e., $E(X) = 0$).

Let $X$ be a discrete random variable with $P(X=1)=p$, $P(X=2)=2p$, and $P(X=3)=1-3p$. Determine all values of $p$ for which this is a valid probability distribution.

The probability mass function of a discrete random variable $X$ is defined by:

$P(X=0)=q$ $P(X=1)=2q$ $P(X=2)=3q$ $P(X=3)=4q$

Find the value of $q$ and hence find $\text{E}(X)$.

Let $X$ be a discrete random variable with probability mass function $P(X=x) = \frac{c}{2^x}$ for $x = 1, 2, 3, 4$.

Determine the value of the constant $c$.