X 0 1 2 3 F(X) 0.5 0.6 0.8 1.0
Finding the Mean of a Discrete Random Variable
The mean, or expected value, of a discrete random variable $X$, denoted as $E[X]$, is calculated by summing the product of each possible value of $X$ and its corresponding probability $P(X=x)$. The formula is:
$E[X] = \sum_{i} x_i P(X=x_i)$
Deriving Probability Mass Function (PMF) from Cumulative Distribution Function (CDF)
The cumulative probability $F(X)$ (or CDF) gives $P(X \le x)$. We need the probability mass function $P(X=x)$ to calculate the mean. We can find $P(X=x)$ using the CDF values:
- $P(X=0) = F(0) = 0.5$
- $P(X=1) = F(1) - F(0) = 0.6 - 0.5 = 0.1$
- $P(X=2) = F(2) - F(1) = 0.8 - 0.6 = 0.2$
- $P(X=3) = F(3) - F(2) = 1.0 - 0.8 = 0.2$
Let's summarize the PMF in a table:
| X | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P(X=x) | 0.5 | 0.1 | 0.2 | 0.2 |
Calculating the Mean Value
Now, we apply the formula for the expected value using the derived PMF:
$E[X] = (0 \times 0.5) + (1 \times 0.1) + (2 \times 0.2) + (3 \times 0.2)$
$E[X] = 0 + 0.1 + 0.4 + 0.6$
$E[X] = 1.1$
The mean value of X is 1.1.
Important Questions from Random Variables
If the odds in favour of any random event A are 5 ∶ 6, then the odds against the event are:
If random variable X follows binomial distribution with parameter n and p with mean 15 and variance 10, then the value of mode is
Let $X$ and $Y$ be continuous random variables with probability density functions $P_X(x)$ and $P_Y(y)$, respectively. Further, let $Y = X^2$ and $P_X(x) = \begin{cases} 1, & x\in (0,1] \\ 0, & \text{otherwise} \end{cases}$
Which one of the following options is correct?- Two fair dice (with faces labeled 1, 2, 3, 4, 5, and 6) are rolled. Let the random variable $X$ denote the sum of the outcomes obtained.
The expectation of $X$ is __________ (rounded off to two decimal places). - Let $X = aZ + b$, where $Z$ is a standard normal random variable, and $a, b$ are two unknown constants. It is given that
$E[X] = 1$, $E[(X – E[X])Z] = –2$, $E[(X – E[X])^2] = 4$,
where $E[X]$ denotes the expectation of random variable $X$. The values of $a, b$ are:
