You can fill up textbooks answering this question so I'm going to go ahead and dedicate my answer to a few inequalities.

Markov's Inequality: if $X$ is any nonnegative integrable random variable and $a > 0$, then

$$\mathbb{P}(X \geq a) \leq \frac{\mathbb{E}(X)}{a}$$

Chebyshev's Inequality: let $X$ be an integrable random variable with finite expected value $\mu$ and finite non-zero variance $\sigma^2$. Then for any real number $k > 0$ then

$$\mathbb{P}(|X-\mu|\leq k\sigma) \geq \frac{1}{k^2}$$

These two inequalities are very powerful as they are true for any distribution that random variable $X$ comes from. For example, if we know that a random variable has mean 0 and standard deviation 1 we know that the probability that this random variable from an unknown distribution has a value between -2 and 2 must be more than .75.

Cramer-Rao Bound: suppose $\theta$ is an unknown deterministic parameter which is to be estimated from measurements $x$. The variance of any unbiased estimator $\hat{\theta}$ of $\theta$ is then bounded by the reciprocal of the Fisher information $I(\theta)$.

$$\mathrm{var}(\hat{\theta})\geq\frac{1}{I(\theta)}$$

This is powerful because if you're unbiased estimated reaches the lower bound then you know your unbiased estimator is the minimum variance unbiased estimator!

Jensen's Inequality: if $X$ is a random variable and $f$ is a convex function, then

$$f\left(\mathbb{E}[X]\right) \leq \mathbb{E}\left[f(X)\right]$$

Like Chebyshev's and Markov's this inequality is applicable all over the place and that's why it's useful!

You can fill up textbooks answering this question so I'm going to go ahead and dedicate my answer to a few inequalities.

Markov's Inequality: if $X$ is any nonnegative integrable random variable and $a > 0$, then

$$\mathbb{P}(X \geq a) \leq \frac{\mathbb{E}(X)}{a}$$

Chebyshev's Inequality: let $X$ be an integrable random variable with finite expected value $\mu$ and finite non-zero variance $\sigma^2$. Then for any real number $k > 0$ then

$$\mathbb{P}(|X-\mu|\leq k\sigma) \geq \frac{1}{k^2}$$

These two inequalities are very powerful as they are true for any distribution that random variable $X$ comes from. For example, if we know that a random variable has mean 0 and standard deviation 1 we know that the probability that this random variable from an unknown distribution has a value between -2 and 2 must be more than .75.

Cramer-Rao Bound: suppose $\theta$ is an unknown deterministic parameter which is to be estimated from measurements $x$. The variance of any unbiased estimator $\hat{\theta}$ of $\theta$ is then bounded by the reciprocal of the Fisher information $I(\theta)$.

$$\mathrm{var}(\hat{\theta})\geq\frac{1}{I(\theta)}$$

This is powerful because if your unbiased estimated reaches the lower bound then you know your unbiased estimator is the minimum variance unbiased estimator!

Jensen's Inequality: if $X$ is a random variable and $f$ is a convex function, then

$$f\left(\mathbb{E}[X]\right) \leq \mathbb{E}\left[f(X)\right]$$

Like Chebyshev's and Markov's this inequality is applicable all over the place and that's why it's useful!

Markov's Inequality and Chebyshev's Inequality are very useful.

Markov's Inequality: if $X$ is any nonnegative integrable random variable and $a > 0$, then

$$\mathbb{P}(X \geq a) \leq \frac{\mathbb{E}(X)}{a}$$

Chebyshev's Inequality: let $X$ be an integrable random variable with finite expected value $\mu$ and finite non-zero variance $\sigma^2$. Then for any real number $k > 0$ then

$$\mathbb{P}(|X-\mu|\leq k\sigma) \geq \frac{1}{k^2}$$

These two inequalities are very powerful as they are true for any distribution that random variable $X$ comes from. For example, if we know that a random variable has mean 0 and standard deviation 1 we know that the probability that this random variable from an unknown distribution has a value between -2 and 2 must be more than .75.

You can fill up textbooks answering this question so I'm going to go ahead and dedicate my answer to a few inequalities.

Markov's Inequality: if $X$ is any nonnegative integrable random variable and $a > 0$, then

$$\mathbb{P}(X \geq a) \leq \frac{\mathbb{E}(X)}{a}$$

Chebyshev's Inequality: let $X$ be an integrable random variable with finite expected value $\mu$ and finite non-zero variance $\sigma^2$. Then for any real number $k > 0$ then

$$\mathbb{P}(|X-\mu|\leq k\sigma) \geq \frac{1}{k^2}$$

These two inequalities are very powerful as they are true for any distribution that random variable $X$ comes from. For example, if we know that a random variable has mean 0 and standard deviation 1 we know that the probability that this random variable from an unknown distribution has a value between -2 and 2 must be more than .75.

Cramer-Rao Bound: suppose $\theta$ is an unknown deterministic parameter which is to be estimated from measurements $x$. The variance of any unbiased estimator $\hat{\theta}$ of $\theta$ is then bounded by the reciprocal of the Fisher information $I(\theta)$.

$$\mathrm{var}(\hat{\theta})\geq\frac{1}{I(\theta)}$$

This is powerful because if you're unbiased estimated reaches the lower bound then you know your unbiased estimator is the minimum variance unbiased estimator!

Jensen's Inequality: if $X$ is a random variable and $f$ is a convex function, then

$$f\left(\mathbb{E}[X]\right) \leq \mathbb{E}\left[f(X)\right]$$

Like Chebyshev's and Markov's this inequality is applicable all over the place and that's why it's useful!