IIT JAM GATE CU SO ASO Statistics
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"Welcome to our statistics channel! 📊 Let's explore the world of numbers together and unlock the power of statistics!"
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Hii, We are excited to announce that we will be initiating topic-wise MCQ tests for entrance and competition exams very soon.
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied.
Chi-square probability curve for degrees of freedom 1 and 2 is -
Equality of several normal population means can be tested by -
Normal distribution, also known as the Gaussian distribution, has several key properties:
1. Symmetry: The normal distribution is symmetric around its mean, meaning that the values on the left and right sides of the mean are equally distributed.
2. Bell-shaped curve: The graph of a normal distribution forms a bell-shaped curve, with the highest point at the mean.
3. Mean, Median, and Mode Equality: In a normal distribution, the mean, median, and mode are equal and located at the center of the distribution.
4. 68-95-99.7 Rule (Empirical Rule): Approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
5. Characterized by Two Parameters: The distribution is completely defined by its mean (μ) and standard deviation (σ).
6. Asymptotic Tail Behavior: The tails of the normal distribution extend infinitely in both directions and never touch the horizontal axis.
7. Standard Normal Distribution: If a normal distribution has a mean of 0 and a standard deviation of 1, it is referred to as a standard normal distribution. Z-scores, representing the number of standard deviations a data point is from the mean, are commonly used in this context.
8. Additivity of Normal Variables: The sum of independent normal random variables is also normally distributed.
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Q. Assume that X Binomial (n, p) for some n≥1 and 0 < p < 1 and Y Poisson (λ) for some λ> 0. Suppose E[X] = E[Y], then
Q. Suppose an unbiased coin is tossed thrice, then what is E(X)? If X denotes number of heads?
✍️Properties of Regression coefficients
👉The correlation coefficient is the geometric mean of the two regression coefficients
👉Regression coefficients are independent of change of origin but not of scale.
👉If one regression coefficient is greater than unit, then the other must be less than unit but not vice versa. ie. both the regression coefficients can be less than unity but both cannot be greater than unity, ie. if b1>1 then b2<1 and if b2>1, then b1<1.
👉Also if one regression coefficient is positive the other must be positive (in this case the correlation coefficient is the positive square root of the product of the two regression coefficients) and if one regression coefficient is negative the other must be negative (in this case the correlation coefficient is the negative square root of the product of the two regression coefficients). ie.if b1>0, then b2>0 and if b1<0, then b2<0
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