$S = \{0, 1, 2, 3\}$, being the number of individuals of a species within a habitat.
Consider the probability distribution of X with $Pr(X = 0) = 0.15$,
$Pr(X = 1) = 0.25$ and $Pr(X = 3) = 0.5$, where Pr denotes probability. The value of
$Pr(X = 2)$ is __________. (Round off to two decimal places)
Understanding the Probability Distribution
We are given a discrete random variable X representing the number of individuals of a species in a habitat. The possible values for X are $S = \{0, 1, 2, 3\}$.
- The probability of X taking the value 0 is given as $Pr(X = 0) = 0.15$.
- The probability of X taking the value 1 is given as $Pr(X = 1) = 0.25$.
- The probability of X taking the value 3 is given as $Pr(X = 3) = 0.5$.
We need to find the probability $Pr(X = 2)$.
Calculating the Missing Probability Pr(X=2)
A fundamental property of any probability distribution is that the sum of probabilities for all possible values of the random variable must equal 1.
Mathematically, this is expressed as:
$ \sum_{x \in S} Pr(X = x) = 1 $Substituting the possible values of X:
$ Pr(X = 0) + Pr(X = 1) + Pr(X = 2) + Pr(X = 3) = 1 $Now, let's plug in the known probability values:
$ 0.15 + 0.25 + Pr(X = 2) + 0.5 = 1 $Combine the known probabilities:
$ 0.90 + Pr(X = 2) = 1 $To find $Pr(X = 2)$, subtract 0.90 from both sides of the equation:
$ Pr(X = 2) = 1 - 0.90 $ $ Pr(X = 2) = 0.10 $The calculated value for $Pr(X = 2)$ is 0.10. This value falls within the specified range of 0.05 to 0.15.
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:
