Question

The probability mass function of X is f(x)= 1/6, x 1.2...,6 then find out the value

Answer 1

For the given probability mass function of X, the mean is 3.5 and the standard deviation is 1.708.

• A discrete random variable X's probability mass function (PMF) is a function over its sample space that estimates the likelihood that X will have a given value. f(x)=P[X=x].
• The total of all potential values for a random variable X, weighted by their relative probabilities, is known as the mean (or expected value E[X]) of that variable.
• Mean(μ) = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6).
• Mean(μ) = (1+2+3+4+5+6)/6
• Mean(μ) = 21/6
• Mean(μ) = 3.5
• The square root of the variance of a random variable, sample, statistical population, data collection, or probability distribution represents its standard deviation. It is denoted by 'σ'.
• A random variable's variance (or Var[X]) is a measurement of the range of potential values. It is, by definition, the squared expectation of the distance between X and μ. It is denoted by 'σ²'.
• σ² = E[X²]−μ²
• σ² = [1²(1/6) + 2²(1/6) + 3²(1/6) + 4²(1/6) + 5²(1/6) + 6²(1/6)] - (3.5)²
• σ² = [(1² + 2²+ 3² + 4²+ 5²+ 6²)/6] - (3.5)²
• σ² = [(1 + 4 + 9 + 16 + 25 + 36)/6] - (3.5)²
• σ² = (91/6) - (3.5)²
• σ² = 15.167-12.25
• σ² = 2.917
• σ = √2.917
• Standard deviation (σ) = 1.708

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