The probability density function of a discrete random variable X is given by the following table: Px(X = 0) = .54 Px(X = 1) = .16 Px(X = 2) = .06 Px(X = 3) = .04 Px(X = 4) = .20 i) Compute E(X). ii) Compute Var(X).
The probability density function of a discrete random variable X is given by the following table: Px(X = 0) = .54 Px(X = 1) = .16 Px(X = 2) = .06 Px(X = 3) = .04 Px(X = 4) = .20 i) Compute E(X). ii) Compute Var(X).
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![**Probability Density Function of a Discrete Random Variable**
The probability density function of a discrete random variable \( X \) is given by the following table:
\[
P_X(X = 0) = 0.54 \\
P_X(X = 1) = 0.16 \\
P_X(X = 2) = 0.06 \\
P_X(X = 3) = 0.04 \\
P_X(X = 4) = 0.20
\]
Tasks:
i) Compute the expected value \( E(X) \).
ii) Compute the variance \( V(X) \).
*Expected Value (E(X)) Calculation:*
The expected value \( E(X) \) of a discrete random variable is given by the formula:
\[ E(X) = \sum [x \cdot P(X = x)] \]
Where \( x \) are the different values that \( X \) can take and \( P(X = x) \) are the corresponding probabilities.
*Variance (V(X)) Calculation:*
The variance \( V(X) \) of a discrete random variable is given by the formula:
\[ V(X) = E(X^2) - [E(X)]^2 \]
Where:
\[ E(X^2) = \sum [x^2 \cdot P(X = x)] \]
These formulas are used to compute the expected value and variance of the given random variable based on its probability distribution.
This information provides essential foundations in statistics for understanding the behavior of a discrete random variable.
For more detailed explanations and examples, please visit our [Educational Materials Section](#).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45fdd75f-924d-4ac4-9d15-03fc1f26bc02%2F9d1a8c3b-238a-49b4-a6d3-fb4fdcbbc3c6%2Ftet4lvi_processed.png&w=3840&q=75)
Transcribed Image Text:**Probability Density Function of a Discrete Random Variable**
The probability density function of a discrete random variable \( X \) is given by the following table:
\[
P_X(X = 0) = 0.54 \\
P_X(X = 1) = 0.16 \\
P_X(X = 2) = 0.06 \\
P_X(X = 3) = 0.04 \\
P_X(X = 4) = 0.20
\]
Tasks:
i) Compute the expected value \( E(X) \).
ii) Compute the variance \( V(X) \).
*Expected Value (E(X)) Calculation:*
The expected value \( E(X) \) of a discrete random variable is given by the formula:
\[ E(X) = \sum [x \cdot P(X = x)] \]
Where \( x \) are the different values that \( X \) can take and \( P(X = x) \) are the corresponding probabilities.
*Variance (V(X)) Calculation:*
The variance \( V(X) \) of a discrete random variable is given by the formula:
\[ V(X) = E(X^2) - [E(X)]^2 \]
Where:
\[ E(X^2) = \sum [x^2 \cdot P(X = x)] \]
These formulas are used to compute the expected value and variance of the given random variable based on its probability distribution.
This information provides essential foundations in statistics for understanding the behavior of a discrete random variable.
For more detailed explanations and examples, please visit our [Educational Materials Section](#).
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