Question

Let \(X\) be \(\chi^{2}(50)\). Approximate \(P(40

Short answer

The approximated probability \(P(40<X<60)\) is 0.6827.

Step-by-step solution

Determine Mean and Variance

The mean and variance of a chi-square distribution is given by its degree of freedom. So, the mean (μ) is 50 and the variance (σ²) is 2*50 = 100. Hence, σ = 10.

Applying Central Limit Theorem

According to the Central Limit Theorem, a chi-square distribution can be approximated to standard normal distribution when the degree of freedom is large (greater than or equal to 30). Hence, normalize the variables using the formula \(Z = (X-\mu)/\sigma\). So the limits 40 and 60 can be normalized as: \(Z1 = (40-50)/10 = -1\) and \(Z2 = (60-50)/10 = 1\).

Computing the Probability

We are now left with the standard normal variable Z with \(P(-1<Z<1)\), which can be determined using a standard normal table or a standard normal calculator. The probability that Z falls between -1 and 1 is approximately 0.6827.